Asymptotic representation theory and the spectrum of a random geometric graph on a compact Lie group

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma(N,L)$ whose vertices are $N$ random points $g_1,\ldots,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma(N,L)$, when $N$ goes to infinity. If $L$ is fixed and $N \to + \infty$ (Gaussian regime), then the largest eigenvalues of $\Gamma(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions. If $L = O(N^{-\frac{1}{\dim G}})$ and $N \to +\infty$ (Poissonian regime), then the random geometric graph $\Gamma(N,L)$ converges in the local Benjamini-Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma(N,L)$. In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl's character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.

(1) If L is fixed and N → +∞ (Gaussian regime), then the largest eigenvalues of Γ geom (N, L) converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions.
(2) If L = O(N − 1 dim G ) and N → +∞ (Poissonian regime), then the geometric graph Γ geom (N, L) converges in the local Benjamini-Schramm sense, which implies the weak convergence in probability of the spectral measure of Γ geom (N, L).
In both situations, the representation theory of the group G provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl's character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of G. In this paper, Z, R, C = R ⊕ iR and H = R ⊕ iR ⊕ jR ⊕ kR = C ⊕ jC denote respectively the set of integers, the field of real numbers, the field of complex numbers, and the division algebra of quaternionic numbers.
1.1. Spectrum of large random graphs. We call graph a pair Γ = (V, E) with V finite set, and E finite subset of the set of pairs {v, w} with v = w in V. In particular, the random graphs that we shall consider in this paper will always be unoriented and simple, that is without loop or multiple edge. Our computations will also involve (deterministic) oriented or labeled graphs, possibly with loops or with multiple edges, but this will be recalled each time by using in particular the terminologies of circuits and reduced circuits (Section 5). The size of a graph is the cardinality N = |V| of its vertex set.
The adjacency matrix of a graph Γ of size N is the matrix A Γ of size N × N, with rows and columns labeled by the vertices of Γ, and with coefficients In particular, the diagonal coefficients of the adjacency matrix of a simple graph are all equal to zero. The adjacency matrix of a graph Γ being a real symmetric matrix, its spectrum Spec(Γ) consists of N real eigenvalues c 1 ≥ c 2 ≥ · · · ≥ c N (see Figure 1). The knowledge of the spectrum yields many informations on the geometry of the graph: mean and maximal number of neighbors of a vertex; chromatic number; number of edges, triangles, spanning trees; expansion properties, Cheeger constant; etc. We refer to [Chu97,GR01] for an introduction to this algebraic graph theory. The purpose of this paper is to study the spectrum of a class of random graphs drawn on certain Riemannian manifolds X, by using the representation theory of the isometry group of X. The simplest example of random graphs that one can think of is when each possible edge {i, j} between points of V = [[1, N]] = {1, 2, . . . , N} is kept at random, according to a Bernoulli law of parameter p, independently for each pair. One obtains the Erdös-Rényi random graphs ( [ER59]), and if p is not too small (e.g. larger than N −1/3 ), the eigenvalue distribution 1 N ∑ N i=1 δ c i of Γ ER (N, p) admits after appropriate renormalisation a deterministic continuous limit, which is the Wigner semicircle law 1 2π √ 4 − x 2 1 −2<x<2 dx; see Figure 2. We refer to the recent papers [EKYY13,EKYY12] for a detailed study of this model, including results on the spacing of eigenvalues and on the edge of the spectrum. 1.2. Random geometric graphs. A more complicated model consists in geometric graphs, that can be defined on any measured metric space. Let (X, d) be a metric space, and m be a Borel probability measure on X. Given a positive real number L > 0, the random geometric graph with N points and level L is the random graph Γ = (V, E) with • the N vertices v 1 , . . . , v N of V chosen randomly in X according to the probability measure m ⊗N on X N .
• an edge between v i and v j if and only if d(v i , v j ) ≤ L.
For the Euclidian case, when X = R p , d(x, y) = x − y is the Euclidian distance and m is a measure on R p , we refer to the monograph [Pen03], where most of the classical questions on random graphs (subgraph counts, threshold for connectivity, existence of a giant connected component, etc.) are answered. In this setting, the spectrum of the adjacency matrix has been studied in [BEJJ06], see also [Bor08,DGK16] for the case where R p is replaced by the torus T p = (R/Z) p . In [DGK16], the distance between points of the torus is the Euclidean distance, whereas in [Bor08] this is the ∞ -distance (which does not correspond to a Riemannian structure). The goal of this article is to extend this work to a general setting of compact Riemannian manifolds that satisfy a certain symmetry property. An important point is that in the regime where L is fixed and N goes to infinity, the asymptotics of the spectrum are discrete instead of continuous. We only get a continuous limiting distribution in the thermodynamic limit where L = L N → 0 is chosen so that the mean number of neighbors of a given vertex is a O(1).
Example 1.1. In Figure 3, we have drawn in stereographic projection a random geometric graph on the real sphere RS 2 , with m equal to Lebesgue's spherical measure, N = 100 points, and L = π 8 (one eighth of the diameter of the space).
FIGURE 3. A random geometric graph on RS 2 , with N = 100 and L = π 8 (the blue circle is the equator).
1.3. Compact symmetric spaces. Let us now explain which measured metric spaces (X, d, m) will be allowed in this paper. We want (X, d) to be a Riemannian manifold (cf. [Jos11]), that is a smooth manifold X endowed with scalar products · | · T x X on each tangent space T x X, these scalar products varying smoothly with x. The Riemannian structure allows one to measure the distance between two points: A geodesic on a Riemannian manifold X is a (smooth) path that minimises locally the distances; it is the solution of an order 2 differential equation, the Euler-Lagrange equation ([Jos11, Lemma 1.4.4]). If X is a compact Riemannian manifold, then for any x ∈ X and any vector v of norm 1 in T x X, there is a unique geodesic γ x,v : R → X with γ x,v (0) = x, γ x,v (0) = v and γ x,v (t) T γ x,v (t) X = 1 for any t ∈ R.
The computation of the spectrum of a random geometric graph on a Riemannian manifold X relies on the harmonic analysis of this space. If X has some symmetry properties, then this harmonic analysis turns into algebraic combinatorics, which allow exact calculations. Thus, in the sequel, we shall restrict ourselves to the more convenient setting of compact symmetric Riemannian manifolds. A compact Riemannian manifold X is called a (globally) symmetric space if, for any x ∈ X, there exists a (unique) involutive isometry s x ∈ Isom(X) that reverses the geodesics, that is to say that it sends γ x,v (t) to γ x,v (−t) for any v ∈ S(T x X). Intuitively, this means that the geodesics meeting at a point x ∈ X are arranged in a nice symmetric way around their starting point, see Figure 4. There is a complete classification non-symmetric space of the (compact) symmetric spaces due to Cartan, see [Hel78,Chapter X]. To simplify a bit the discussion, we shall assume X to be simply connected. In the general case, a connected but non simply connected compact symmetric space X admits a universal cover X which is still a compact symmetric space, and whose covering map π : X → X has finite degree. This allows one to transfer most results and techniques from X to X. Later, we shall for instance explain how to deal with the case of the special orthogonal groups SO(n ≥ 3), which are symmetric spaces with fundamental group π 1 (SO(n)) = Z/2Z, and which are covered by the simply connected spin groups Spin(n).
If X is a simply connected compact symmetric space, then it is isometric to a unique product X 1 × X 2 × · · · × X r of such spaces, with each X i that cannot be split further. The X i 's are called irreducible or simple simply connected compact symmetric spaces. Then, the classification of simple and simply connected compact symmetric spaces (in short, ssccss) is the following: (1) either X = G is one of the classical simple and simply connected compact Lie groups, associated to the root systems of type A n≥1 , B n≥2 , C n≥3 , D n≥4 , G 2 , F 4 , E 6 , E 7 or E 8 . In this case, (Isom(X)) 0 = G × G.
(2) or, X = G/K, with G simple and simply connected compact Lie group, and K closed subgroup with G θ,0 ⊂ K ⊂ G θ , where G θ denotes the set of fixed points of an involutive automorphism θ : G → G, and G θ,0 is the connected component of the neutral element in G θ . In this case, (Isom(X)) 0 = G.
We call a ssccss of group type or of non-group type according to the aforementioned classification. The Riemannian structure on each ssccss X = G or X = G/K is unique up to a scalar multiple, and we shall explain in a moment how to construct it. Moreover, this Riemannian structure on X yields a natural volume form dω with finite mass. After renormalisation, this volume form produces a probability measure m = dω/( X dω) on X that is invariant by the group of isometries of X. Therefore, every ssccss is naturally endowed with a distance d and a probability measure m.
Our objective is to study random geometric graphs in the general setting of ssccss. The harmonic analysis of the two types (group and non-group) is in theory quite similar, and in each case there is an explicit description of the spherical functions of the space (see the works of Helgason [Hel70,Hel78,Hel84]). However, the manipulation of the spherical functions in the non-group case (which includes Grassmannian manifolds and Lagrangian Grassmannian manifolds) is in practice more difficult. Therefore, in this paper, we shall in many cases restrict our study to the group type, hence to the classical sscc Lie groups. We shall only treat the non-group type when the results extend almost immediately to this case. More precisely, the non-group type ssccss will appear in the following sections: • Section 3.4: when studying the Gaussian regime in the symmetric spaces with rank one, the irreducible characters are replaced by the zonal spherical functions, which are in this case the orthogonal Laguerre or Jacobi polynomials, hence explicit and easy to manipulate.
• Section 4: the Benjamini-Schramm local convergence holds for all the symmetric spaces, and the argument is exactly the same in the group and non-group case.
1.4. Compact Lie groups and normalisation of the Riemannian structure. In the following we fix a simple simply connected compact (in short sscc) Lie group G. Given a compact Lie group G, the tangent space g = T e G G at the neutral element e G is endowed with a structure of Lie algebra; we denote [X, Y] = (ad X)(Y). The opposite of the Killing form is a symmetric and positive-definite bilinear form on g which is invariant by the adjoint action of G on g. We transport this scalar product to any tangent space T g G by the rule where L g −1 : G → G is the multiplication on the left by g −1 , and d g L g −1 is the differential of this map at g. By construction, the Riemannian structure thus obtained is G-invariant on the left, and it is also G-invariant on the right since · | · g is Ad(G)-invariant. In the sequel, the Riemannian structure on a classical sscc Lie group will always be the one associated to the bilinear form of Equation (1). The corresponding balls B (x,L) = {y ∈ G | d(x, y) ≤ L} for the geodesic distance d will be described in Section 3.1. We shall recall in a moment that almost all the sscc Lie groups are classical groups of matrices over the real, the complex or the quaternionic numbers. The Killing form writes then as tr(ad X • ad Y) = c Re(tr(XY)), with c = n − 2 when g = so(n); c = 2n when g = su(n); and c = 4n + 4 when g = sp(n) (the real part of the trace is only needed in this last case). For the corresponding Riemannian structure, the distance in SU(2) between two unitary matrices U 1 and U 2 is d(U 1 , U 2 ) = 2 √ 2 |θ(U 1 U −1 2 )|, where e ±iθ(M) are the two eigenvalues of a unitary matrix M ∈ SU(2), with θ(M) ∈ [0, π]. Indeed, d(U 1 , U 2 ) = d(U 1 U −1 2 , I 2 ) = d(diag(e iθ(U 1 U −1 2 ) , e −iθ(U 1 U −1 2 ) ), I 2 ), and a geodesic connecting I 2 to the diagonal matrix diag(e iθ , e −iθ ) is which has constant speed γ (t) = 2 √ 2 θ. In particular, the diameter of SU(2) with this normalisation is 2 √ 2 π.
If X = G/K is a ssccss of non-group type, we denote π X : G → X the canonical projection, and o = π X (e G ) = K. The tangent space T o X identifies through T e G π X with the Killing orthogonal complement p of the Lie subalgebra k of K in g. The restriction of the scalar product from Equation (1) to p can be transported to any tangent space T x X by using the action of G: , where x = g · o and A g : X → X is the action of g on X. By construction, the Riemannian structure thus obtained makes G act on X by isometries. In the sequel, the Riemannian structure on a ssccss of non-group type will be the one obtained by this construction. However, in the specific case of ssccss of rank one (Section 3.4), we shall multiply this Riemannian metric by a multiplicative constant so as to fit the classical definitions. The following example explains why this modification is natural.
Example 1.3. Suppose that X = CP n = SU(n + 1)/S(U(n) × U(1)) is the complex projective space. If Z = (z 0 , . . . , z n ) belongs to C n+1 \ {0}, we denote [Z] = [z 0 , z 1 , . . . , z n ] the corresponding line in CP n . The reference point in X is o = [0, . . . , 0, 1]. The standard Riemannian metric on CP n is the Fubini-Study metric, defined by where Z 2 = ∑ n i=0 |z i | 2 and (Y, Z) is the real scalar product on C n corresponding to this norm. In this formula, a vector V ∈ C n+1 is sent to the element of T [Z] CP n which is the derivative at t = 0 of the smooth curve ([Z + tV] = [z 0 + tv 0 , . . . , z n + tv n ]) t∈R ; the kernel of the linear map C n+1 → T [Z] CP n is the line [Z]. In particular, if [Z] = o, then we have the identification C n = T o CP n , and the scalar product on C n inherited from the Fubini-Study metric is simply V | V = ∑ n−1 i=0 |V i | 2 . Now, the Riemannian structure obtained by SU(n + 1)-transport of the restriction to p = T o CP n of the opposite Killing form is a scalar multiple of this metric. Indeed, we have , and the tangent map T I n+1 π CP n sends the skew-Hermitian matrix M(z 0 , . . . , z n−1 ) to the vector (z 0 , . . . , z n−1 ) in C n = T o CP n . As the Killing form of SU(n + 1) is (A, B) → (2n + 2) tr(AB), we conclude that the scalar product on C n given by the structure of symmetric space is V | V = (4n + 4) ∑ n−1 i=0 |V i | 2 , hence (4n + 4) times the "standard" scalar product.
To conclude this section, let us detail a bit more the classification of sscc Lie groups. They are: • the special unitary groups SU(n) with n ≥ 2: • the compact symplectic groups USp(n) with n ≥ 2: where (M † ) ij = M ji , the conjugate of a quaternionic number a + ib + jc + kd being a − ib − jc − kd; • the spin groups Spin(n) with n ≥ 7, which are double covers of the special orthogonal groups and which are simply connected (whereas π 1 (SO(n)) = Z/2Z for any n ≥ 3).
There are also 5 exceptional cases which are associated to the root systems G 2 , F 4 , E 6 , E 7 and E 8 , and which all related to the geometry of the algebra of octonions (see [Bae02]). For instance, consider the exceptional Jordan algebra A(3, O) (the so-called Albert algebra), which is the algebra of real dimension 27 that consists in Hermitian 3 × 3 octonionic matrices, endowed with the Jordan product One can show that the automorphism group of this algebra is a simply connected simple compact Lie group of real dimension 52, associated to the root system F 4 . As the exceptional Lie groups do not possess adequate systems of (matrix) coordinates, it is quite difficult to express distances on them. Thus, in these cases, our theoretical results will remain mainly abstract. On the other hand, for the "classical" sscc groups SU(n ≥ 2), USp(n ≥ 2), Spin(n ≥ 7), all our results will be explicit; see the appendix (Section 6) for explanations and computations on these groups. Note that one can extend many of our results to a slightly more general setting, with reductive connected Lie groups instead of sscc Lie groups. The case of the special orthogonal groups SO(n), which are not simply connected, is for instance explained in Remark 6.1.
1.5. Main results and outline of the paper. When studying the random geometric graphs Γ = Γ geom (N, L) on a compact Riemannian manifold X, there are two interesting asymptotic regimes which one can consider: (1) the Gaussian regime, where L is fixed but N goes to infinity; in this setting the adjacency matrix A Γ is dense.
(2) the Poissonian regime, where L = L N decreases to zero in such a way that each vertex of Γ has a O(1) number of vertices; in this setting the adjacency matrix A Γ is sparse.
Gaussian regime. The adjacency matrix A Γ can be considered as a finite-dimensional (random) approximation of the operator of convolution by the kernel h(x, y) = 1 d(x,y)≤L . In particular, a result due to Giné and Koltchinskii [GK00] relates the asymptotics of the spectrum of A Γ to the eigenvalues of the operator of convolution by h (Section 2.1). Suppose that X = G is a ssccss of group type. By using the representation theory of compact Lie groups, one can compute these eigenvalues, which drive the highest frequencies of the random geometric graph, that is the asymptotic behavior of the largest eigenvalues of A Γ geom (N,L) . In Sections 2.2 and 2.3, we present the arguments from representation theory that show that there is one limiting eigenvalue c λ of for each dominant weight λ of the group G. This eigenvalue c λ has a multiplicity related to the dimension of the corresponding irreducible representation of G. In Section 3, we complete this theoretical result by an explicit calculation of c λ (Theorem 3.1). Thus, each limiting eigenvalue c λ is given by a finite linear combination of values of Bessel functions of the first kind J β , see Section 3.2. In Section 3.3, we deduce from this result an estimate of the spectral radius and of the spectral gap of the matrix A Γ geom (N,L) when L is fixed and N goes to infinity. If instead of a group G we consider a ssccss of non-group type G/K, the same techniques apply in theory, but with the irreducible representations replaced by the spherical representations of the pair (G, K), and the irreducible characters by the zonal spherical functions. These functions can be cumbersome to deal with in the general case, but if G/K has rank one (meaning that there are no totally geodesic flat submanifold of dimension strictly larger than 1), then they are simply the Laguerre or Jacobi polynomials, and the computations can be explicitly performed; we explain this in Section 3.4.
Poissonian regime. We consider again a general ssccss X. The connection distance L N is normalised as follows: with > 0 fixed. Then, the number of neighbors of any vertex v i of Γ geom (N, L N ) follows a binomial law where vol(X) is the volume of the symmetric space X for the volume form associated to the Riemannian structure given by Equation (1), and vol(B(v i , L N )) is the volume of the ball in X with center v i and radius L N . As N goes to infinity, this volume behaves like the volume of a Euclidean ball with the same dimension, which is Γ(1 + dim X 2 ) Therefore, in the limit N → ∞, the number of neighbors of any vertex v i of Γ geom (N, L N ) has a law close to a Poisson law of parameter c(dim X) vol(X) ; in particular it is a O(1). More generally, for any n ≥ 1 and any fixed vertex v i , one can show that the subgraph of Γ geom (N, L N ) which consists in vertices at distance smaller than n from v i has a limit in law in the set of rooted finite graphs. This is the convergence in the local Benjamini-Schramm sense [BS01], and the limit only depends on the dimension dim X and on the parameter /vol(X); see Sections 4.1-4.3, and in particular our Theorem 4.6. We prove this result by developing a general theory relating the convergence of pointed metric spaces to the local Benjamini-Schramm convergence of the random geometric graphs drawn on such spaces; see Theorem 4.11.
It is then known from [BL10, ATV11, BLS11, Bor16] that under appropriate assumptions, the local convergence of random graphs implies the convergence in law of the spectral measures of the graphs Γ geom (N, L N ) towards a limiting probability measure µ. We check the conditions to apply this result in Section 4.4; one has in particular to verify that two random roots in Γ geom (N, L N ) give rise to two independent local limits, and this is a consequence of the structure of group or homogeneous (symmetric) space. We also prove that the limiting measure µ of the spectral measures ν N is determined by its moments, and that we have convergence in probability of the moments.
From random graphs to a conjecture in representation theory. Moving on from there, one can try to obtain more information on the limiting distribution µ. For instance, one expects it to be compactly supported, but this result does not follow from the abstract link between local Benjamini-Schramm convergence and convergence of the spectral measure.
The crude upper bounds proving that µ is determined by its moments also do not imply the compactness of the support. This leads one to try to improve these bounds, and to develop techniques that enable one to compute all the moments M s = R x s µ(dx). In the sequel, we focus on the case where X = G is a sscc Lie group.
(1) In Section 5.1, we start by giving a circuit expansion of the expected moments, which is a combinatorial expansion of E[ R x s ν N (dx)] involving certain labeled graphs. This expansion implies that each moment M s is a polynomial with degree s − 1 in the parameter (see Theorem 5.4).
(2) Since the graph limit in the local sense does not depend on the group G and only depends on dim G and /vol(G), the same is true for the limiting spectral measure µ, and therefore one can replace G by a simpler group, namely, the torus T dim G . We plan to pursue this approach in a forthcoming paper; even with this simplification, it is not easy to obtain good bounds on the moments M s , as it amounts to count (reduced) circuits with certain weights (see Remark 5.10).
Aside from the search for good upper bounds on the moments M s , there is actually an interest in keeping the base model G instead of the flat model T dim G . It turns out that the Poissonian regime of random geometric graphs, which we approach in Section 4 with the geometric notion of local Benjamini-Schramm convergence, can also be studied with representation theoretic tools (Section 5). In this setting, the computation of the moments sheds a different light on the degeneration from the Gaussian to the Poissonian regime, and it eventually leads to an algebraic conjecture which we state below, and which concerns certain joint integrals of characters of G. Let us explain briefly how one is led to it: (1) The formulas in the Gaussian regime (Section 3) rely mainly on the Weyl formula for the characters of the irreducible representations of G. When going from the Gaussian to the Poissonian regime and trying to compute the first moments M s in the model G (specifically, for s ≤ 5), the Weyl formula degenerates into a product of partial derivatives, and the sums over dominant weights become integrals over Weyl chambers and products thereof; see Section 5.2. This is a typical result from asymptotic representation theory, and as far as we know this explicit degeneration has not been pointed at previously in a study of random objects associated to groups.
(2) The previous degeneration concerns the terms of the circuit expansion of a moment M s which corresponds to a reduced circuit with one vertex. When s ∈ {6, 7}, one starts to see contributions from reduced circuits with two vertices, and their asymptotics is related to asymptotic formulas for the Littlewood-Richardson coefficients associated to large dominant weights. In the general case, these asymptotic formulas come from the Kashiwara-Lusztig theory of crystal bases and the Berenstein-Zelevinsky theory of string polytopes; they involve positive measures with piecewise polynomial densities, against which one integrates partial derivatives of Bessel functions in order to compute the contributions of the reduced circuits on two vertices; see Section 5.3. A similar kind of degeneration has been observed when studying Brownian motions in Weyl chambers, see [BBO05].
(3) Starting with s ≥ 8, the circuit expansion of M s involves some reduced circuits with more than 3 vertices. These contributions are limits of certain series whose terms involve graph functionals of the irreducible characters of G (Section 5.4). If we suppose that the limiting process happens in the same way as for 2-vertices reduced circuits, then we obtain the following conjecture. Suppose that G is a sscc Lie group and that S = (V(S), E(S)) is a finite graph, possibly with multiple edges or loops and with an arbitrary orientation a → b of each edge {a, b} ∈ E(S). We associate to each edge e ∈ E(S) a dominant weight λ e , which parametrises an irreducible finitedimensional representation of G; see Section 2.3 for a reminder on this theory. The graph functional associated to G, S and to this choice of dominant weights is: where V(S) = {1, 2, . . . , k}, dg is the Haar measure on G, and ch λ e is the character of the irreducible representation V λ e with highest weight λ e . When S has one or two vertices and several edges or loops, one recovers classical quantities such as the dimensions dim V λ or the Littlewood-Richardson coefficients c λ,µ ν . The graph functionals defined by Equation (2) are generalisations of these quantities, and thus it is natural to try to compute them. Our study of random geometric graphs in the Poissonian regime led us to the following conjecture, which seems important: Conjecture 1.4. Fix a sscc Lie group G and a connected graph S as above, with k vertices and r edges. We denote ZΩ the weight lattice of G, see Section 2.3. There exists a sublattice A S ⊂ (ZΩ) r with maximal rank rd and such that: • If the integrality condition (λ e ) e∈E(S) ∈ A S is not satisfied, then GF S ((λ e ) e∈E(S) ) vanishes.
• If the integrality condition (λ e ) e∈E(S) ∈ A S is satisfied, then GF S ((λ e ) e∈E(S) ) equals the number of integer points in a polytope P((λ e ) e∈E(S) ) whose generic dimension is Here by generic we mean that the dimension of the polytope is equal to the right-hand side as soon as the dominant weights λ e are in the interior of the Weyl chamber. The equations that determine the polytope P((λ e ) e∈E(S) ) are affine functions of the weights, and P((λ e ) e∈E(S) ) is a part of the string cone SC (G r ) of the sscc Lie group G r .
This conjecture implies some vanishing results which do not seem trivial at all; see Remark 5.9. We probably would never have obtained this conjecture without examining this concrete problem of computation of the moments M s ; it is a typical example of the interplay between random objects considered on spaces which admit a group of symmetry, and the asymptotic representation theory of these groups. Note that the conjecture is interesting in itself, but not at all for the original problem stated at the beginning of the paragraph (computing bounds on M s ), which is more of a combinatorial nature and which we do not intend to solve here (it is then required to consider the flat model T dim G ). Our last Section 5 is devoted to the presentation of this conjecture, following the arguments that we have briefly exposed above. We also found it essential to explain how the degeneration from the Gaussian to the Poissonian regime of geometric graphs can be followed in representation theoretic terms, with degenerations of the Weyl formula, of sums over dominant weights and of Littlewood-Richardson coefficients; these results will certainly be interesting for specialists of asymptotic representation theory. A reader with a probabilistic background might not be familiar with the arguments from representation theory. He will find in this case: • in Section 2, a reminder of the classical Cartan-Weyl representation theory of sscc Lie groups; we also use this section to fix notations.
• an appendix (Section 6) with a list of conventions and results (choice of the maximal tori, description of the root systems and of the weight lattices, computation of the volumes of the groups, etc.); it allows one to apply concretely our results to the classical sscc Lie groups (SU(n), USp(n), SO(n) and Spin(n)).
• a second appendix (Section 7) with an explanation of the theory of crystals of representations and string polytopes; one can skip these explanations if one is not interested in the algebraic details that leads to Conjecture 1.4.
The reader with a more advanced knowledge of these algebraic results can safely skip these sections. kernel random matrices at University Paris-Sud (Orsay), who introduced me to the problem of the spectrum of random geometric graphs. In particular, I am much indebted to Édouard Maurel-Segala, who explained to me the somewhat easier case of geometric graphs on ∞tori. I also thank Reda Chhaibi for several discussions that we had on the subject, and for his precious comments. I learned about the link between Benjamini-Schramm convergence and the convergence of spectral measures from talks given in the Workgroup on random matrices and graphs at the Institut Henri Poincaré (MEGA), and I am very thankful to its organizers. A significant progress on this project was made during a conference in Les Diablerets (Switzerland) in January 2017, and I would like to thank the organizers of this conference for their invitation. Finally, I am indebted to an anonymous referee for many constructive remarks on a previous version of this paper, which allowed to improve a lot the presentation of our results.

INGREDIENTS FROM REPRESENTATION THEORY
In this section, G is a fixed sscc Lie group, and L > 0 is a fixed level. The uniform probability measure m on this space (Haar measure) will be denoted dg or dx. By combining a result of Giné and Koltchinskii and the representation theory of compact groups, we relate the spectrum of the random adjacency matrix A(N, L) of Γ geom (N, L) to the spectrum of an integral operator on L 2 (G, dg). This integral operator will be explicitly diagonalised in Section 3. The present section will also allow us to introduce many ingredients from representation theory that we shall use throughout the paper.
2.1. The Giné-Koltchinskii law of large numbers. We denote L 2 (G, dg) the set of complexvalued measurable and square-integrable functions on G. Let h(x, y) be a real symmetric function on G, such that G 2 (h(x, y)) 2 dx dy < +∞. The convolution by h induces a integral operator T h on L 2 (G, dg): This operator is auto-adjoint and of Hilbert-Schmidt class: given any (countable) orthonormal basis (e i ) i∈I of L 2 (G, dg), ( T h HS ) 2 = ∑ i∈I ( T h (e i ) L 2 (G) ) 2 = ( h L 2 (G 2 ) ) 2 . Therefore, T h is a compact operator, and it admits a discrete real spectrum, which we label by integers: with lim |k|→∞ c k = 0 (here, we add an infinity of zeroes to the sequence (c k ) k∈Z if needed, for instance when T h is of finite rank). The Hilbert-Schmidt class ensures that ∑ k∈Z (c k ) 2 < +∞. Now, a general result due to Giné and Kolchinskii (see [GK00]) ensures that one can approximate the operator T h by the random matrices where the v i 's are independent random variables chosen according to the Haar measure dg on G. The spectrum of T h (N) is a random set which approximates Spec(T h ) in the following sense: Theorem 2.1 (Giné-Koltchinskii, Theorem 3.1 in [GK00]). Under the previous assumptions, This result yields readily the asymptotics of the spectrum of A(N, L) when L is fixed and N goes to infinity. Indeed, Now, notice that the operator T h is in fact an operator of convolution by a function of one variable: for f ∈ L 2 (G, dg), where Z L (g) = 1 d(g,e G )≤L . Here we used the invariance of the distance d by the action of the group G. Hence, T h is an operator of convolution on L 2 (G, dg) by a function in L 2 (G, dg) which is invariant by conjugation. The next paragraphs explain how to use the representation theory of G in order to compute the eigenvalues of such a convolution operator (and therefore, the asymptotics of Spec(Γ geom (N, L)) in the regime where L is fixed and N → +∞).

Convolution on a compact Lie group.
Let G be a compact topological group endowed with its Haar measure dg. We denote G the set of classes of isomorphism of irreducible finite-dimensional complex representations of G; it is always countable, and for any element λ ∈ G corresponding to a representation (V λ , ρ λ : G → GL(V λ )), one can find an Hermitian scalar product · | · V λ on V λ which is invariant by G. This scalar product induces an adjunction u → u * on End(V λ ), and we then endow The basic theorem which allows to understand convolution in L 2 (G, dg) is: Theorem 2.2 (Peter-Weyl, 1927). For λ ∈ G and f ∈ L 2 (G, dg), denote the Fourier transform of f . The map f → f from L 2 (G) to L 2 ( G) = ⊥ λ∈ G End(V λ ) is an isometry of Hilbert spaces and an isomorphism of algebras (with L 2 (G, dg) endowed with the convolution product).
We refer to [Bum13, Chapter 4] for a proof of this important result. It implies that the eigenspaces for the convolution on the left by Z L correspond via the Fourier transform to subspaces of the endomorphism spaces End(V λ ), that are eigenspaces for the multiplication on the left by Z L (λ). Moreover, as Z L is invariant by conjugation, the convolution on the left by Z L is the same as the convolution on the right by Z L . In the Fourier world, this means that each endomorphism Z L (λ) is in the center of End(V λ ), hence a scalar matrix c λ id V λ . Therefore, the eigenspaces for the convolution on the left by Z L are exactly the spaces End(V λ ), and the corresponding eigenvalues are the where χ λ (g) = tr(ρ λ (g)) d λ is the normalised character of the irreducible representation λ. Thus, to summarise: Proposition 2.3. Denote Z L (g) = 1 d(g,e G )≤L with G sscc Lie group. The eigenvalues of the operator on L 2 (G) of convolution on the left or on the right by Z L are in bijection with the irreducible representations λ ∈ G. Each eigenvalue c λ has multiplicity (d λ ) 2 and is given by the formula c λ = G Z L (g) χ λ (g) dg.
The next paragraph will allow us to identify the set G, and to compute the dimensions d λ . Proposition 2.3 extends readily to the case of symmetric spaces X = G/K, see Section 3.4.
Remark 2.4. In the following, we denote ch λ the non-normalised character tr ρ λ . A direct consequence of the Peter-Weyl theorem 2.2 is that the collection of non-normalised characters (ch λ ) λ∈ G forms an orthonormal basis of L 2 (G) G , the space of square-integrable and conjugacy-invariant functions on G. Moreover, one has the convolution rule ch λ * ch µ = δ λ,µ d λ ch λ .

Weight lattice and combinatorics of the highest frequencies.
When G is a (semi)simple simply connected compact Lie group, the set G is classically described by Weyl's highest weight theorem, see for instance [GW09, Theorems 3.2.5 and 3.2.6]. Let T be a maximal torus in G, and ZΩ be the lattice of weights, a weight of G being a character ω : T → U(1) = {z ∈ C | |z| = 1} such that there exists a unitary representation (V, ρ) of G with The weights form a free module over Z for the operation of pointwise product. A standard convention is to denote additively the composition law in ZΩ, and to write evaluations of weights as t → e ω (t) (instead of ω(t)). Let g C be the complexification of the Lie algebra g of G, and t C the complexification of the Lie algebra t of T. The map ω → T e G (e ω ) allows one to see the weights as elements of t * C . The dual of the Killing form restricted to RΩ = R ⊗ Z ZΩ is positive-definite. Hence, one has a natural scalar product · | · on the lattice of weights, which can be shown to be W-invariant, where W = Norm(T)/T is the Weyl group. We decompose the roots of G (non-zero weights of the adjoint representation of G on g C ) in two disjoint sets Φ + and Φ − = −Φ + of positive and negative roots; then, and on the other hand, the positive roots determine a cone in RΩ which is a fundamental domain for the action of the Weyl group (the Weyl chamber). The intersection of the two aforementioned sets is then in bijection with G: Theorem 2.5 (Weyl, 1925). An irreducible unitary representation (V, ρ) of G admits a unique highest weight λ, which is maximal with respect to the partial order on weights induced by the cone C. This highest weight has multiplicity one and enables one to reconstruct the irreducible representation (V, ρ). Moreover, λ is an arbitrary dominant weight in C ∩ ZΩ, so The dimension of the representation V λ with highest weight λ is given by the formula Example 2.6. Suppose G = SU(3). A maximal torus is The lattice of weights ZΩ is spanned by the two fundamental weights e ω 1 (t) = t 1 and e ω 2 (t) = (t 3 ) −1 . The positive roots are e α 1 (t) = t 1 (t 2 ) −1 , e α 2 (t) = t 2 (t 3 ) −1 and e α 1 +α 2 (t) = t 1 (t 3 ) −1 . The dominant weights, which label the irreducible representations of SU(3), are the linear combinations n 1 ω 1 + n 2 ω 2 with n 1 , n 2 ∈ N; on Figure 5, they correspond to the dots that are included in the cone C. The dimension of V λ with λ = n 1 ω 1 + n 2 ω 2 is d λ = (n 1 +1)(n 2 +1)(n 1 +n 2 +2) 2 . For instance, the adjoint representation of SU(3) on sl(3, C) has highest weight λ = ω 1 + ω 2 , and dimension 8. If one replaces the coordinates (n 1 , n 2 ) by the integer partition λ = (λ 1 ≥ λ 2 ≥ λ 3 ) with λ 1 = n 1 + n 2 , λ 2 = n 2 and λ 3 = 0, one gets the classical formula which generalises to higher dimensions.
Corollary 2.7. Let Γ = Γ geom (N, L) be a random geometric graph of fixed level L on a sscc Lie group G, and A(N, L) be its adjacency matrix. In the sense of Theorem 2.1, the limit of Spec( consists of one limiting eigenvalue c λ for each dominant weight λ ∈ C ∩ ZΩ. The multiplicity of c λ is and the value of c λ = G Z L (g) χ λ (g) dg will be given in Theorem 3.1.
In the appendix (Section 6), we give for each classical case (unitary groups, compact symplectic groups, spin groups): • a maximal torus T; • the corresponding weight lattice ZΩ and the root system Φ; This allows one to make explicit Corollary 2.7 and all the forthcoming theorems. In the examples hereafter, we shall focus on the groups SU(2) and SU(3). For SU(2), the weight lattice is drawn in Figure 6, and it is one-dimensional; many intuitions come from a detailed study of this toy-model. Remark 2.8. Corollary 2.7 generalises readily to more general compact Lie groups, by replacing the set of dominant weights C ∩ ZΩ by an adequate sublattice of it. In particular, one can treat without additional work the case of the unitary groups U(n), which are not simple, since they have a non-trivial center; and the case of the special orthogonal groups SO(n), which are simple Lie groups but are not simply connected. In the appendix we detail this last case, where ZΩ is replaced by an index 2 sublattice (see Remark 6.1). Thus, though we shall not mention it again hereafter, every result obtained in the sequel whose statement starts by "Given a sscc Lie group. . . " also holds mutatis mutandis for the non-sscc but classical Lie groups U(n) and SO(n).

ASYMPTOTICS OF THE SPECTRUM IN THE GAUSSIAN REGIME
In this section, we compute the limiting eigenvalues c λ introduced in Corollary 2.7. We obtain a formula which involves Bessel functions of the first kind and an alternate sum over elements of the Weyl group, see Theorem 3.1. These computations allow one for instance to estimate the spectral radius and the spectral gap of a random geometric graph Γ geom (N, L) with fixed level L; see Section 3.3. On the other hand, we shall see in Section 5 that the alternate sums involved in the formula for c λ degenerate in the Poissonian regime into certain partial derivatives. Therefore, the calculation of the eigenvalues c λ will be useful for studying both asymptotic regimes (Gaussian and Poissonian). In Section 3.4, we also explain how to extend our results to ssccss of non-group type; the computations become explicit in rank one and they involve Laguerre of Jacobi orthogonal polynomials.
3.1. Distances on a compact Lie group. Since c λ = G 1 d(g,e G )≤L χ λ (g) dg, we need to explain how to deal with distances on a sscc compact Lie group G. We fix as before a maximal torus T ⊂ G, and we denote t ⊂ g the corresponding Lie subalgebra. Every element g ∈ G is conjugated to an element t ∈ T, which is unique up to action of the Weyl group W. Consequently, as Z L is a function invariant by conjugation, in order to compute the function Z L (g) = 1 d(g,e G )≤L , it suffices to know its values on T. Now, the maximal torus is a totally geodesic flat submanifold of G, and the exponential map exp : t → T is locally isometric from a neighborhood of 0 to a neighborhood of e G . In all the classical cases, the injectivity radius of the exponential map is at least equal to π (this is clear from the description of the maximal tori given in Section 6). This enables one to reduce the calculation of c λ to an integration over a ball in the Euclidean space t. Indeed, by Weyl's integration formula (see [Bum13,Chapters 17 and 22]), since Z L is invariant by conjugation, • dt is the uniform probability over the torus T; • ∆(t) = ∑ w∈W ε(w) e ρ (w(t)); • for any w ∈ W viewed as an element of SO(RΩ), ε(w) = (−1) (w) is the determinant of the transformation w, or equivalently the parity of the number of reflections with respect to the walls of the Weyl chamber that are needed to write w.
Suppose L < π. Then, the integral can be taken over t instead of T: Indeed, the probability measure dt corresponds via the exponential map to the rescaled Lebesgue measure where dX is the volume form on t which is associated to the Riemannian structure given by · | · g ; and 2πt Z is the kernel of the exponential map exp : t → T, and a lattice with maximal rank in t. In the classical cases, the volumes vol(t/t Z ) are computed in Section 6.6.

Asymptotics of the largest eigenvalues.
In the sequel we always denote d = dim T = rank G the rank of the group G; in geometric terms, it is the dimension of a totally geodesic flat submanifold, and for a compact Lie group this is the dimension of a maximal torus. In the classical cases, we have rank(SU(n)) = n − 1, and rank(Spin(2n)) = rank(Spin(2n + 1)) = rank(USp(n)) = n. If λ is the highest weight of an irreducible representation V λ , then the restriction of the corresponding character to the torus T is given by Weyl's formula Notice that the denominator in Weyl's character formula is the quantity ∆(t) previously introduced. Therefore, in Equation (3), writing |∆(t)| 2 = ∆(t) ∆(t) makes ∆(t) appear in the numerator and the denominator. We can simplify it to get: As the measure dX is invariant by W, one can gather the terms of the double sum according to the value w = w 2 w −1 1 , and one obtains: X∈t, X g ≤L e (λ+ρ−w(ρ))(X) dX.
Each integral is a value of the Fourier transform of the unit ball in R d , that is a value of a Bessel function of the first kind. Indeed, recall that if B d is the unit ball in R d , we have where J β is the Bessel function of the first kind of index β, defined by the power series Given a weight lattice RΩ of a sscc Lie group with rank d = rank(G), it is convenient to introduce the modified Bessel function which is a W-invariant analytic function on RΩ. Then, In this formula, the modified Bessel function involves the norm λ + ρ − w(ρ) , which is the norm of the weight lattice introduced in Section 2.3, and which is computed in the appendix for the classical cases. We have finally shown: Theorem 3.1. Suppose that the level L is smaller than π. Let λ be a highest weight in G. The eigenvalue c λ is given by the following formula: where d = rank(G) is the dimension of a maximal torus T ⊂ G, and J RΩ is the modified Bessel function on the weight space RΩ.
Example 3.2. Consider G = SU(2). Its weight lattice ZΩ is spanned by the fundamental weight e ω (diag(e iθ , e −iθ )) = e iθ . The norm of a weight kω is |k| , and on the other hand, Therefore, for k ≥ 1 and L < π, The multiplicity of the eigenvalue c k is equal to (k + 1) 2 for any k ≥ 0.
The formula for c λ with λ = n 1 ω 1 + n 2 ω 2 dominant weight in the Weyl chamber involves 6 weights close to λ, namely, all the weights λ + µ with µ ∈ {0, 3ω 1 , 3ω 2 , 2ω 2 − ω 1 , 2ω 1 − ω 2 , 2ω 1 + 2ω 2 }, see Figure 8. Thus, and each eigenvalue c n 1 ,n 2 has multiplicity m n 1 ,n 2 = ( (n 1 +1)(n 2 +1)(n 1 +n 2 +2) 2 ) 2 . In this formula, the norm of a weight k 1 ω 1 + k 2 ω 2 is 3.3. Spectral radius and spectral gap. One thing that is not entirely clear from Theorem 3.1 is that the largest eigenvalues c λ correspond roughly to the smallest dominant weights λ in the Weyl chamber C. This is not a perfect correspondence: for instance, when G = SU(2), the dominant weights kω with k ≥ 0 yields constants c k = c kω whose modules are not strictly decreasing with k. However, the two largest eigenvalues in this case are always c 0 and c 1 , see the discussion later in this paragraph. One thing that is always true and easy to prove is that the largest eigenvalue corresponds to the zero weight: Proposition 3.4. For any L < π, the eigenvalue c λ with the largest absolute value is obtained when λ = 0 is the trivial weight. Hence, the spectral radius of the graph Γ geom (N, L) is asymptotically equivalent to Proof. The eigenvalue c λ is given by the integral c λ = G Z L (g) χ λ (g) dg, with Z L (g) nonnegative function, and χ λ renormalised character that has always its module smaller than 1. The maximum value is obtained when χ λ (g) = 1 for every g, that is for the trivial representation of G.
Example 3.5. When G = SU(2), it is easy to prove that the two largest eigenvalues c k are always where sinc(x) = sin x x . Indeed, L being fixed, the eigenvalue c k is proportional to the function . For any value of L ∈ (0, π), the function g L looks like the one of Figure 9, and the two values of g L at k = 0 and k = 1 always fall on the first decreasing section of the curve. Hence, they yield the asymptotic spectral gap of a random geometric graph Γ geom (N, L) on SU(2), with L fixed and N going to infinity. In general, a level L being fixed, the map λ → c λ is proportional to In the definition of g L , the alternate sum of modified Bessel functions is a discretisation of the partial derivative ((∏ α∈Φ + ∂ α ) J β )(Lx); see Section 5.2, where this argument will be made rigorous for the Poisson regime. The discrete partial derivative g L can be extended to the whole Weyl chamber C, and it is then an oscillating function that goes to 0 as the norm of its parameter grows to infinity, in a fashion very similar to what happens for SU(2). We have drawn in Figure 10 a function g L for L > 0 and the group G = SU(3); here the oscillations are very small and barely visible. In this case, it is clear that the non-zero weights that yield the largest eigenvalues are ω 1 and ω 2 , which correspond to the fundamental representations of the group.  The only thing that might prevent one of the fundamental representations of the group to provide the second largest eigenvalue is if L is too large, forcing the points of the lattice L(ZΩ) that are neighbors of the weight 0 to be at the bottom of the first oscillation of g L . This forbids us to state a universal theorem for the spectral gap, though we are also unable to provide a counterexample. In practice, a level L being fixed, one can use the asymptotic behavior of the Bessel functions to get rid of the points of the lattice L(ZΩ) that are too far from 0, and then there is only a finite number of values of g L to examine in order to determine the spectral gap. Thus, in most cases, the spectral gap of Γ geom (N, L) is asymptotically equivalent to where ω 1 is the fundamental weight: • that corresponds in the classical cases to the geometric representation of the group of matrices, • and that maximises g L most of the time.
3.4. Extension to symmetric spaces with rank one. In this paragraph, we explain how to adapt the arguments of the two previous sections to the case of a ssccss X = G/K of nongroup type. Roughly speaking, all the theoretical arguments from Section 2 adapt readily by replacing the irreducible representations of G by the spherical representations of the pair (G, K); on the other hand, the concrete computations from this section can be performed without too much additional work if the space G/K has rank one, because in this case the zonal spherical functions are polynomials of one parameter.

Spherical representations and zonal functions.
In the sequel, we fix a compact symmetric space X = G/K, and we denote dx the unique G-invariant probability measure on X, which is the image of the Haar measure by the canonical projection from G to X. We call spherical an irreducible representation V λ of G which admits a non-zero K-fixed vector, so is not reduced to {0}. It can be shown that (V λ ) K has then dimension 1, and also that the subset G K ⊂ G of spherical representations is the intersection of the Weyl chamber C with a sublattice of the lattice of weights ZΩ; this is the Cartan-Helgason theorem, see [Hel84, Chapter V, Theorem 4.1], as well as [Sug62] and [GW09, Section 12.3]. Later, we shall only be interested in the case of compact symmetric spaces with rank one, in which case this sublattice has also rank one and will be explicitly described by Proposition 3.8. Given a spherical representation V λ with λ ∈ G K , we fix a spherical vector e λ in (V λ ) K with e λ e λ V λ = 1. The vector e λ is unique up to multiplication by a complex number with modulus 1. The (normalised) zonal spherical function on X associated to the spherical representation V λ is The spherical transform of a bi-K-invariant function is defined for f bi-K-invariant and λ ∈ G K by We endow the space L 2 ( G K , d • ) = ⊥ λ∈ G K C with the coordinatewise product and with the Hilbert structure coming from the scalar product where on the right-hand side we have the usual scalar product on C. The analogue of Theorem 2.2 in this setting is: is an isometry of Hilbert spaces and an isomorphism of commutative algebras. Moreover, if c ∈ L 2 (K\G/K, dg), then the convolution on the right is a Hilbert-Schmidt operator; its eigenvalues c sph (λ) are in correspondence with the spherical weights λ ∈ G K , each c sph (λ) having multiplicity d λ .
A reformulation of the first part of this theorem is that the zonal spherical functions form an orthogonal basis of L 2 (K\G/K, dg), with the convolution rule zon λ * zon µ = δ λ,µ d λ zon λ . We refer to [Hel84, Chapter V] for a proof of this result and a study of the spherical functions of a compact symmetric space; an analogous treatment for finite Gelfand pairs is provided by [CSST08, Chapter 4], and the whole discussion from loc. cit. adapts readily to compact symmetric spaces by replacing the finite sums by integrals against Haar measures. Now, in the setting of random geometric graphs with fixed level L on a symmetric space of nongroup type, the Giné-Koltchinskii theorem still applies, so the limit in the sense of Theorem 2.1 of the spectrum of is the spectrum of the integral operator with h(x, y) = 1 d(x,y)≤L . This operator writes as the right-convolution R c with c(g) = Z L (g) = 1 d(gK,K)≤L bi-K-invariant function on G. Consequently, the analogue in the setting of ssccss of non-group type of Proposition 2.3 is: Proposition 3.7. Denote Z L (g) = 1 d(gK,K)≤L with X = G/K ssccss of non-group type. The eigenvalues of the operator on L 2 (X) of convolution on the right by Z L are in bijection with the spherical representations λ ∈ G K . Each eigenvalue c λ has multiplicity d λ and is given by the Symmetric spaces with rank one. The abstract result from Proposition 3.7 still holds if X is connected but not necessarily simply connected, so in the sequel of this subsection we remove this assumption. Then, the zonal integrals from the previous proposition can be computed when X has rank one, which is equivalent to one of the following assertions: • X is a compact symmetric space with rank one, meaning that X does not contain a totally geodesic flat submanifold with dimension at least 2; • X is a 2-point homogeneous compact connected Riemannian manifold: if x 1 , y 1 , x 2 , y 2 ∈ X satisfy d(x 1 , y 1 ) = d(x 2 , y 2 ), then there exists an isometry i : X → X such that i(x 1 ) = x 2 and i(y 1 ) = y 2 ; • X is one of the following spaces: the real spheres RS n≥1 = SO(n + 1)/SO(n); the real, complex and quaternionic projective spaces HP n≥2 = USp(n + 1)/(USp(n) × USp(1)); and the exceptional octonionic projective plane OP 2 = F 4 /Spin(9).
We refer to [Wol67, Chapter 8] for a proof of the equivalence between the two first assertions; the classification and the harmonic analysis of these spaces can be found in [Gri83,AH10] and [VV09, Chapter 3]. All these spaces are simply connected but the one-dimensional sphere RS 1 = T and the real projective spaces RP n≥2 , which are twofold-covered by the real spheres RS n . In the following, we endow RS n (respectively, KP n with K ∈ {R, C, H}) with its usual Euclidean coordinates (x 1 , x 2 , . . . , x n+1 ) with ∑ n+1 i=1 (x i ) 2 = 1 (respectively, with its usual homogeneous coordinates [x 1 : x 2 : · · · : x n+1 ]). For the exceptional space OP 2 , we have to be careful because of the non-associativity of the product of octonions, but there is a dense affine chart of OP 2 whose points [θ 1 : θ 2 : 1] are labeled by pairs (θ 1 , θ 2 ) of octonions, such that one can manipulate these homogeneous coordinates in exactly the same way as for the other projective spaces. In this affine chart, we set θ 3 = 1; in general, a set of homogeneous coordinates [θ 1 : θ 2 : θ 3 ] is allowed for a point in OP 2 if the algebra spanned by the three octonions θ 1 , θ 2 , θ 3 is associative, see [Joh76,Ada96,Bae02] for details.
This formula differs by a multiplicative constant from the canonical Riemannian metric on the symmetric space G/K which we detailed in Section 1.4. On the other hand, with the same choice of coordinates, a bi-K-invariant function on the group G, which is a K-invariant function on the symmetric space X, is also a function of this single parameter Consequently, the integrals from Proposition 3.7 are integrals over one single real parameter in [−1, 1] (real spheres) or in [0, 1] (projective spaces), whose distribution under the Haar measure is: In the case of a projective space KP n , one obtains a β-distribution with parameters a and b which depends on the field K of the projective space and on the rank n. The distribution of the spherical coordinate x on the real sphere RS n will be denoted By the remark stated just after the Cartan analogue 3.6 of the Peter-Weyl theorem, the zonal spherical functions zon λ with λ ∈ G K form an orthogonal basis of the space of Kinvariant functions on X, with the normalisation condition zon λ (e G ) = 1. On the other hand, the orthogonal polynomials with respect to the distribution θ n (ds) or β a,b (dx) form an orthogonal basis of the space of functions of the parameter x or s which are squareintegrable. Since the bi-K-invariant functions are functions of this single parameter, the two orthogonal bases correspond, and the following proposition describes these functions and the associated spherical representations.
Proposition 3.8. Consider a symmetric space X = G/K with rank one. There is in each case a dominant weight ω 0 ∈ C ∩ ZΩ such that G K = Nω 0 : The corresponding zonal spherical functions are the Legendre polynomials in the case of real spheres and the Jacobi polynomials both formulas being instances of Rodrigues' formula for orthogonal polynomials.
The first part of the proposition is an immediate application of the Cartan-Helgason theorem which identifies the spherical dominant weights; the second part is treated in [AH10, Chapter 2] in the case of spheres, and in [VV09] for the other spaces.
Asymptotics of the largest eigenvalues. By combining Proposition 3.7 and the explicit formula for zonal spherical functions from Proposition 3.8, we can now compute the limiting eigenvalues of N . Let us for instance treat the case of projective spaces. We have for k ≥ 1: For k = 0, the spherical representation is the trivial one and c 0 is the mean of the function 1 d(x,b)≤L , hence the normalised volume in X of this ball. The computations are analogous in the case of real spheres, with Laguerre polynomials instead of Jacobi polynomials. We conclude: Theorem 3.9. Let X be a sscc with rank one, and L a level in (0, π 2 ). In the sense of Theorem 2.1, the limit of Spec( OP 2 (sin L) 16 ( 1+8 cos 2 L +36 cos 4 L+120 cos 6 L ) 165 (cos L) 8 (sin L) 16 J (5,9),k−1 (cos 2 L) In particular, one can as in Section 3.3 use these formulas in order to compute the asymptotic spectral radius and spectral gap of Γ geom (N, L) in these cases.
Thus, up to the multiplicative factor sin 2 L 4 , all the limiting eigenvalues of the random geometric graph of level L can be obtained by looking at the values at x = cos L of the family of functions see Figure 11.
. The limiting eigenvalues of a random geometric graph on the 2dimensional real sphere in the Gaussian regime.

ASYMPTOTICS OF THE GRAPH AND OF ITS SPECTRUM IN THE POISSONIAN REGIME
In this section, we fix a ssccss X (of group or non-group type), and we are interested in the asymptotic behavior of the spectrum of Γ geom (N, L N ) when N goes to infinity and L N goes to 0 in the following prescribed way: with > 0 fixed. As explained in the introduction, with this normalisation of L N , the expected number of neighbors of a fixed vertex of Γ geom (N, where vol(X) is the volume of the space X, and c(dim X) is the volume of a Euclidean unit ball in R dim X . When X = G is a Lie group, its volume vol(G) is computed in Section 6.6; for the other cases, we refer to [AY97]. We now set where c 1 (N) ≥ c 2 (N) ≥ · · · ≥ c N (N) are the eigenvalues of the adjacency matrix of Γ geom (N, L N ). For each N, ν N is a random element of M 1 (R), the set of Borel probability measures on the real line. The remainder of this article focuses on studying the asymptotic behavior of the random spectral measures ν N . We shall in particular prove that there exists a probability measure µ ∈ M 1 (R) which depends only on , vol(X) and dim X, and such that ν N N→+∞ µ, (5) where denotes the convergence in law; see Theorem 4.20. In Equation (5), the convergence occurs in probability; this makes sense since M 1 (R) is a polish space for the topology of weak convergence, so in particular it is metrisable; see [Bil99, Chapter 1].
There are at least two possible approaches in order to prove the convergence in law (5) for the most general result, which relies on arguments from the theory of von Neumann algebras. We recall briefly this theory in Section 4.1. In the setting of Poissonian random geometric graphs: (i) We have a random point process (v n ) n∈N (the random vertices) which takes place on a space which is locally almost isometric to an Euclidean vector space.
(ii) As n goes to infinity, this random point process has locally almost the same statistics as a Poisson point process.
(iii) The geometric graph built from this random point process converges then in the local Benjamini-Schramm sense, and this implies the weak convergence of the spectral measures.
In the almost correspondences listed above, the geometric graphs can be locally modified with a positive probability, so we have to be very careful if we want to prove rigorously the local convergence of our random geometric graphs. To this purpose, we solve a more general problem by giving a sufficient condition for a convergent random point process on a convergent sequence of metric spaces to give rise to a sequence of random graphs which is locally convergent (Theorem 4.11). In Section 4.2, we recall the notion of pointed Lipschitz convergence for proper metric spaces, and we present a similar notion of convergence for proper metric spaces endowed with a random point process. In Section 4.3, we relate these notions of convergence to the local convergence of random geometric graphs under a mild regularity hypothesis. Our result implies in particular the Benjamini-Schramm convergence of the Poissonian random geometric graphs on a compact connected symmetric space X, the limit being the geometric graph drawn from a Poisson point process on R dim X (Theorem 4.6). This geometric argument combined with the aforementioned result from [Bor16] implies the convergence of the spectral measures ν N towards some probability measure µ (see Theorem 4.20). For this result, we shall use in addition to the previous argument the fact that in a Poissonian random geometric graph, the neighborhoods of two vertices which are at macroscopic distance are asymptotically independent when N goes to infinity.
• Method of moments (Section 5). Another more naive approach is to try to compute the moments of the measure ν N , and to prove that they all converge in probability towards the moments of a measure which is determined by its moments. We shall prove at the end of Section 4.4 that the limiting measure µ is indeed determined by its moments. Section 5 proposes then a combinatorial method in order to compute these limiting moments, and it explains how the computation of these moments is related to the asymptotic representation theory of the Lie group G. As detailed in the introduction, we do not solve entirely the problem of the computation of the moments of µ, but this alternative approach leads quite surprisingly to a general conjecture on certain functionals of the irreducible representations of the group.
4.1. Benjamini-Schramm local convergence and continuity of the spectral map. The notion of local convergence of random graphs concerns random rooted graphs. Since all the vertices of a random geometric graph play the same role, looking at rooted graphs (Γ, r) instead of simple graphs Γ will not be a problem hereafter. We denote G • the set of all connected locally finite rooted graphs (Γ, r) = (V, E, r): is a simple graph, with V possibly infinite but countable; • r ∈ V is a distinguished vertex and all the vertices of Γ are connected to v by a finite path; We identify two connected locally finite rooted graphs (Γ 1 , It should not be confused with the geodesic distance d if the vertices of Γ are points in a Riemannian manifold X. We have a natural homomorphism of rooted graphs where Γ(r, n) is the subgraph of Γ whose vertices are the v's in V such that d Γ (r, v) ≤ n, and whose edges are those of Γ that connect vertices v, w such that d Γ (r, v) ≤ n and d Γ (r, w) ≤ n. For instance, if (Γ, r) is the lattice Z 2 rooted at the origin r = (0, 0), then We endow G • with the following distance: It is known that (G • , d • ) is a complete separable metric space. Moreover, the topology corresponding to d • is the projective limit of the discrete topologies on the sets G • (n): Let us now introduce randomness in this framework. Since G • is a polish space, the space M 1 (G • ) of Borel probability measures on the space of rooted graphs is again a polish space. We say that a sequence of random rooted graphs (Γ N , r N ) N∈N converges in the local Benjamini-Schramm sense towards a random rooted graph (Γ, r) if the probability distributions L (Γ N ,r N ) and L (Γ,r) of these random rooted graphs satisfy We have the following characterisation of the local Benjamini-Schramm convergence: Proposition 4.1. A sequence of random rooted graphs (Γ N , r N ) N∈N converges in the local sense if and only if, for any n ∈ N and any rooted finite graph γ n ∈ G • (n), This equivalence is stated without proof at the beginning of [BS01]; it is relatively easy to prove once one remarks that any open subset of G • is a finite or countable disjoint union of open balls.
If Γ is a finite graph on N vertices, its spectral measure ν Γ is 1 . This definition can be extended to certain infinite (random) rooted graphs as follows. Given (Γ, r) = (V, E, r) ∈ G • , we can consider the adjacency operator where 2 c (V) is the space of finitely supported functions on V, which is dense in 2 (V). This operator is self-adjoint, and it admits at least one self-adjoint extension to 2 (V). If Γ has a uniformly bounded degree, then the self-adjoint extension is unique and it is a continuous linear operator 2 (V) → 2 (V). However, in general, there might be several different selfadjoint extensions of A Γ , and these extensions can be unbounded operators. We say that the graph Γ or its adjacency operator A Γ is essentially self-adjoint if the self-adjoint extension A Γ : 2 (V) → 2 (V) is unique. In this case, given a root r of Γ, we define the spectral measure µ (Γ,r) of the rooted graph by the following formula: where C + denotes the upper half-plane. The measure µ (Γ,r) is a Borel probability measure in M 1 (R), and its existence and unicity is obtained by using Herglotz's representation theorem of holomorphic functions on the upper half-plane, and the standard properties of the resolvent of a self-adjoint (possibly unbounded) linear operator. We refer to [Sch12] for the spectral theory of unbounded operators.
Remark 4.2. In this general setting, we cannot a priori use the moments M s≥1 = 1 r | (A Γ ) s (1 r ) in order to define µ (Γ,r) . Indeed, without additional assumptions, these quantities might correspond to several different probability measures; see however the end of Section 4.4.
Given a distribution L ∈ M 1 ess (G • ) ⊂ M 1 (G • ) supported by essentially self-adjoint rooted graphs, we can finally define its spectral measure µ L by µ L = E L [µ (Γ,r) ]. The process of taking the expectation of a random probability measure is what one expects: for any bounded continuous function . This formula defines a positive linear functional µ L on C b (R), and by [Lan93, Chapter IX, §2, Theorem 2.3] this functional is uniquely determined by a Borel probability measure µ L in M 1 (R). We have thus defined a spectral map This construction extends the notion of spectral measure of a finite graph. Indeed, given a finite graph Γ, let us denote U(Γ) the uniformly pointed graph constructed from Γ: it is the random connected finite graph (Γ, r) with r uniformly chosen among the vertices of Γ, and where we only keep the connected component of the root r. This measure is supported by connected finite graphs, which are of course essentially self-adjoint. An easy computation shows then that the measure µ U(Γ) defined above is simply equal to ν Γ ; see e.g. [Bor16, beginning of Section 2.3].
Remark 4.3. Above and also in the sequel, the spectral measures of finite graphs are denoted by the letter ν, whereas the expected spectral measures of random essentially self-adjoint rooted graphs are denoted by the letter µ. The reader should pay attention to the fact that in the first case the spectral measure ν can be random if the graph is random, whereas the notation µ is always used for deterministic measures.
As far as we know, it is unknown whether the spectral map µ is continuous on the whole space M 1 ess (G • ) of essentially self-adjoint random rooted graphs, but the restriction to a smaller subspace is known to be continuous.
Here, G •• is endowed with the smallest topology which makes the two projections (Γ, r 1 , r 2 ) → (Γ, r 1 ) and (Γ, r 1 , r 2 ) → (Γ, r 2 ) continuous towards G • . For any finite graph Γ, U(Γ) is unimodular, and conversely, a unimodular distribution L of random rooted graphs which is supported by connected finite graphs is necessarily a mixture of uniformly pointed graphs U(Γ). It was shown by Benjamini and Schramm that the unimodular distributions form Theorem 4.4. If (Γ, r) is a random rooted graph chosen according to a unimodular distribution L, then Γ is L-almost surely essentially self-adjoint. In other words, Then, the restriction of the spectral map µ to M 1 uni (G • ) is continuous with respect to the Benjamini-Schramm local convergence and to the weak convergence of measures.
These facts are proven in [Bor16, Proposition 2.2]. They imply in particular that if (Γ N ) N∈N is a sequence of random graphs such that U(Γ N ) → L for some L in M 1 (G • ) (and in fact in Then, there is a simple criterion which allows one to get rid of the expectation and to obtain the convergence in probability of the spectral measures: Proposition 4.5. Given a finite graph Γ, we denote U 2 (Γ) the law in M 1 (G •• ) of a random birooted graph (Γ, r 1 , r 2 ) with (r 1 , r 2 ) chosen uniformly among the pairs of vertices of Γ. Let (Γ N ) N∈N be a sequence of random finite graphs such that U 2 (Γ N ) → L ⊗ L, with L ∈ M 1 (G • ). Note that this implies in particular the local convergence U(Γ N ) → L. Then, ν Γ N N→∞ µ L in probability.
Proof. See [BL10, Theorem 1] or [BLS11, Corollary 12]; the arguments in loc. cit. are used in a less general setting, but they hold without modification. 4.2. Pointed Lipschitz and random pointed Lipschitz convergence. In the remainder of this section, we fix a ssccss X, a parameter > 0, we consider the sequence of random geometric graphs Γ N = Γ geom (N, L N ), with L N as in Equation (4). We denote U(Γ N ) = (Γ N , r N ) with r N uniformly chosen among the vertices of Γ N , and where it is understood that we then only look at the connected component of this root r N . We recall that ν N is the (random) spectral measure of the random geometric graph Γ geom (N, L N ), and we shall also denote µ N = E[ν N ]; with the notations previously introduced, µ N = µ U(Γ geom (N,L N )) . The discussion of the previous section shows that the convergence in probability ν N µ and the deterministic convergence µ N µ are quite close results, and that the second (weaker) result is an immediate consequence of: Theorem 4.6 (Local convergence).
(1) The sequence (Γ N , r N ) N∈N converges in the local Benjamini-Schramm sense towards an infinite random rooted graph (Γ ∞ , r). As a consequence, there exists a Borel probability measure µ on R such that µ N N→∞ µ.
(2) The limit (Γ ∞ , r) has the following distribution. We consider a Poisson point process P on R dim X with intensity vol(X) Leb, where Leb is the standard Lebesgue measure. We take r = 0 and we connect points of P {0} when their Euclidean distance is smaller than 1. Then, Γ ∞ has the distribution of the connected component of the root vertex {0}. In particular, the local limit depends only on dim X and on the parameter vol(X) .
Remark 4.7. The arguments used hereafter adapt readily to any connected compact homogeneous Riemannian manifold X = G/H; in particular, since we do not use any argument from representation theory in this section, the assumption of simple connectedness on the symmetric spaces is here superfluous.
Let us give an intuitive explanation of Theorem 4.6. When looking at the s-neighborhood (in the sense of graph distance) of a random root r N in Γ geom (N, L N ), this s-neighborhood only depends on what happens in a small ball of radius O(sL N ) around r N , and this ball is almost isometric to its Euclidean counterpart in R dim X . Then, the restriction of the point process {v 1 , . . . , v N } to the small ball converges towards a Poisson point process, since each v i has a probability O( 1 N ) to be in the small ball, and since there are N independent random points v 1 , . . . , v N . Theorem 4.6 is therefore a natural result, but let us insist on the fact that its rigorous proof cannot be made short, for the following reason. Since we only have a quasiisometry between the small ball B X in X and its tangent projection B Euclidean in R dim X : The projection in B Euclidean of a geometric graph with level L N in B X is not a geometric graph with level L N in B Euclidean .
Therefore, we need to be very careful with the various approximations involved in the previous intuitive explanation. To overcome the aforementioned difficulties, we shall see Theorem 4.6 as a particular case of a more general result, which states roughly that if a sequence of pointed metric spaces converges in a suitable way, and if one chooses random points on these spaces in a way that is also convergent, then the corresponding geometric graphs converge under adequate assumptions in the local Benjamini-Schramm sense. The existence of such a result is not really surprising, but we could not find in the literature a set of sufficient conditions for the local convergence of graphs in this setting. The remainder of this subsection is devoted to the introduction of all the required hypotheses. In Section 4.3, we shall then show that these hypotheses lead to the aforementioned connection between convergence of metric spaces and convergence of random geometric graphs (Theorem 4.11). We shall also prove in this paragraph and in the next one that all the required hypotheses for Theorem 4.11 are fulfilled in the Poissonian regime of random geometric graphs on a ssccss, leading to a proof of Theorem 4.6.
We start by recalling the notion of pointed Lipschitz convergence [Gro07]. Given two compact metric spaces (X, d) and (X , d ), we say that they are Lipschitz equivalent if there exists an homeomorphism f : X → X such that with c and C strictly positive constants. The Lipschitz distance between two compact metric spaces is then defined by where dil( f ) denotes the dilation constant of an homeomorphism, defined by We It is easy to adapt the definition of the Lipschitz distance to pointed compact metric spaces: if (X, x, d) and (X , x , d ) are two pointed compact metric spaces (compact metric spaces with a distinguished point), we define their pointed Lipschitz distance by This metric yields a topology on the set CMS • of pointed isometry classes of pointed compact metric spaces. Next, we consider pointed proper metric spaces, that is to say metric spaces (X, d) with a distinguished point x and such that every closed ball B X (x, R) with R ≥ 0 is compact. Note that these hypotheses imply that every closed ball in X is compact. We denote PMS • the set of pointed isometry classes of such spaces. For every R ≥ 0, we have a natural map and these maps allow one to endow PMS • with the topology of pointed Lipschitz convergence: a sequence of pointed proper metric spaces (X N , • π R ((X, x, d)) for any R ≥ 0. This is the adequate definition that we shall use hereafter for convergence of metric spaces. In this setting, we endow M Radon (X) with the * -weak topology: a sequence of positive Radon measures (µ n ) n∈N converges towards a Radon measure µ if, for any φ ∈ C c (X), µ n (φ) → µ(φ). This topology is also called the vague topology, and we refer to [Bou81, Chapter 3] for a detailed study of it. With respect to the σ-field spanned by the vague topology, for any Borel subset A ⊂ X, the map (1) We have In particular, this implies that (X N , x N , d N ) → (X, x, d) in the pointed Lipschitz topology.
(2) Consider a continuous function φ : X → R which is compactly supported on a ball Beware that the second condition is weaker than the statement ( f N,R ) ((M N ) |B X N (x N ,R) ) M |B X (x,R) (in law and for the vague topology), because we only allow test functions φ that vanish on the boundary (and outside) of B X (x, R).
Proposition 4.9. Let X be a ssccss. We denote d the geodesic distance on X with the normalisation given by Equation (1); t N = N 1 dim X and d N = t N d; o = e G the neutral element if X = G is a group, and o = π X (e G ) the reference point if X = G/K is not a group. We denote M N the point process on X obtained by taking N independent points at random according to the Haar measure. As N goes to infinity, where d T o X is the Euclidean distance on T o X associated to the opposite Killing form or its restriction, and P (λ) is the Poisson point process on T o X whose intensity is λ Leb, Leb being the Lebesgue measure associated to the distance d T o X . The convergence in Equation (6) is in the random pointed Lipschitz sense.
Proof. In the sequel, since we shall consider families of distances on X, in order to avoid any ambiguity, we shall indicate the distance d of a ball B(x, r) = B (X,d) (x, r) in X. We also denote x = T o X, which is the Lie algebra of X if X is a group, and a subspace of the Lie algebra of G if X = G/K is of non-group type. Fix R > 0. By the aforementioned result from [Gro07, Proposition 3.15], there exist some bijective maps f N,R : which are smooth, which send the reference point o to 0, and such that Let φ be a bounded measurable function compactly supported on B x (0, R). If P = P ( 1 vol(X) ) is the Poisson process on x with intensity 1 vol(X) Leb = dt vol(X) , then the Laplace transform of P(φ) is given by the Campbell formula: On the other hand, the Laplace transform of (( f N, Since f N,R is a smooth quasi-isometry, by the change of variables formula, the image by the map f N,R of the restriction of the Haar measure to the ball where m N,R (t) is a smooth positive function that converges uniformly to 1 on B x (0, R). Then, the change of variables This ensures the convergence in law of the restricted point processes, hence the convergence in the random pointed Lipschitz sense. Notice that, since the previous convergence holds for any bounded measurable function φ, given a family (φ 1 , . . . , φ s ) of bounded measurable functions compactly supported on B x (0, R), we also have  Note that the regularity assumption is equivalent to the fact that almost surely, no pair of points (y, z) of {x} {atoms of M} are exactly at distance L for a fixed positive real number L. Let us first see why this general result implies Theorem 4.6: Proof of Theorem 4.6. With t N = N 1 dim X , we already know that (X, r N , t N d, M N ) N∈N converges in PMS •, towards The differences between this statement and Proposition 4.9 are the following: • We have replaced the reference point o by a random atom r N of M N , which is added to M N−1 .
• We place ourselves on (R dim X , 0, d Euclidean ) instead of (T o X, 0, d T o X ).
However, the second modification amounts to an isometry between x = T o X and R dim X , whereas the first point is clearly solved by using the transitive action of the compact Lie group G associated to X. Now, the limiting space is obviously regular with respect to random geometric graphs, because given L > 0, there is almost surely no atom of the Poisson point process exactly at distance L from another atom, or at distance L from 0. Therefore, we have the local convergence By scaling, the left-hand side is also Γ geom (X, r N , d, M N , L N ) = (Γ N , r N ), whereas the righthand side has the same law as Γ geom R dim X , 0, d Euclidean , δ 0 + P vol(X) , 1 .
We now turn to the proof of Theorem 4.11, which we split in several lemmas. A first consequence of the regularity assumption is that the random increasing map r → M(B X (x, r)) is almost surely continuous at r = R, for any fixed radius R. Indeed, this amounts to the almost sure continuity of the map l → card(π 1 (Γ geom (X, x, d, M, l))) at l = R. A generalisation of this property will be stated in Lemma 4.13. A less trivial consequence of the assumptions of Theorem 4.11 is the following: Lemma 4.12. Suppose that the atoms of M + δ x are simple, and consider (X N , x N , d N , M N ) N∈N sequence in PMS •, that converges in the random pointed Lipschitz sense to (X, x, d, M). Then, each of the random point processes M N + δ x N and M + δ x is uniformly separated: for any R > 0 and any ε > 0, there exists η > 0 and an integer N 0 = N 0 such that ∀N ≥ N 0 , P[M N + δ x N has two atoms in B X N (x N , R) at distance smaller than η] ≤ ε; P[M + δ x has two atoms in B X (x, R) at distance smaller than η] ≤ ε.
Proof. In B X (x, R), we fix a finite sequence (y k ) 1≤k≤K that is η-dense: B X (x, R) ⊂ K k=1 B X (y k , η). This is possible since B X (x, R) is compact. We set φ k (t) = (1 − d(y k ,t) 4η ) + . Note then that if two points p 1 and p 2 are at distance smaller than 3η 2 , then there is at least one y k such that d(p 1 , y k ) ≤ η and d(p 2 , y k ) ≤ 5η 2 , and therefore such that Conversely, if an atomic measure P satisfies P(φ k ) ≥ 9 8 , then there are at least two atoms p 1 and p 2 of P in the support of φ k , and therefore at distance d(p 1 , p 2 ) ≤ 8η. We have thus shown, for any atomic measure P: the minimal distance between atoms of P |B X (x,R) is less than 3η 2 ⇒ (P(φ 1 ), . . . , P(φ K )) belongs to the closed set K k=1 (R + ) k−1 × 9 8 , +∞ × (R + ) K−k ⇒ the minimal distance between atoms of P |B X (x,R+4η) is less than 8η .
The parameters R and ε being fixed, the probability that M + δ x has two atoms in B X (x, R + 4η) at distance smaller than 8η goes to 0 as η goes to 0, because M + δ x is supposed without multiplicity (we also use the fact that the random point processes that we are studying are assumed to be locally finite). So, one can find η such that P the minimal distance between atoms of (M + δ x ) |B X (x,R+4η) is less than 8η ≤ ε 2 .
A fortiori, We introduce the maps f N,R+4η which are almost isometries between the balls B X N (x N , R + 4η) and B X (x, R + 4η). By assumption, Therefore, for N large enough, these probabilities are smaller than ε, and this implies: R+4η) ) at distance less than 3η 2 ≤ ε.
However, for N large enough, f N,R+4η modifies the distances by a factor smaller than 3 2 , therefore, P there are two atoms of (M N + δ x N ) |B X N (x N ,R+4η) at distance less than η ≤ ε.
This clearly implies the result.
A similar result that we shall use later is a property of uniform continuity of the maps R → M(B X (x, R)) and R → M(B X N (x N , R)): Lemma 4.13. For any R > 0 and ε > 0, there exists η > 0 and an integer N 0 such that The proof of this second lemma is entirely similar to the one of Lemma 4.12, and relies on the use of adequate test functions. In the sequel, we fix a rooted finite graph γ n ∈ G • (n), and ε > 0. The symbols Γ N and Γ stand for Γ geom (X N , x N , d N , M N , L) and Γ geom (X, x, d, M, L); in the sequel we shall deal with numerous approximations of these random graphs. Note that if R ≥ nL, then the structure of π n (Γ, x) only depends on the restriction of the point process M to the ball B X (x, R). We fix R ≥ nL + 4, and then η < min 1, L 4 sufficiently small such that with probability at least 1 − ε, • the random rooted graphs π n (Γ geom (X, x, d, M, L + cη), x) with c ∈ [−4, 4] are all the same (regularity condition); • the atoms of (M + δ x ) |B X (x,R+η) are all separated by strictly more than 2η (Lemma 4.12); We denote A ε the event corresponding to these three conditions; P[A ε ] ≥ 1 − ε. If η is sufficiently small and N 0 is sufficiently large, then for N ≥ N 0 , on an event A N,ε with probability larger than 1 − ε, we also have the same two last conditions satisfied by M N in X N : • the atoms of (M N + δ x N ) |B X N (x N ,R+η) are all separated by strictly more than 2η; • the cardinality (M N + δ x N )(B X N (x N , R − η)) is the same as (M N + δ x N )(B X N (x N , R + η)). We now proceed to a kind of discretisation of the random geometric graph Γ N . We fix a set partition Ψ = Ψ 1 Ψ 2 · · · Ψ of the ball B X (x, R) such that diam(Ψ l ) ≤ η 2 for any l ∈ [[1, ]], and we set Π N, is almost an isometry. If N is taken large enough, then f N,R+η modifies the distance between two points p 1 and p 2 by a factor c = c(p 1 , p 2 ) with Therefore, Π N is a set partition with and such that diam(Π N,l ) ≤ η for any l ∈ [[1, ]]. If we place ourselves on the event A N,ε , then (M N + δ x N )(Π N,l ) ≤ 1 for any l ∈ [[1, ]], because otherwise (M N + δ x N ) |B X N (x N ,R+η) would have two atoms at distance smaller than η. Hence, we have fixed for any N ≥ N 0 a grid Π N with arbitrary small size and such that, with very high probability, the atoms of (M N + δ x N ) |Π N fall into the cases of this grid with at most one atom in each case. In the following, we use the same notation Π N for the set partition (Π N,1 , . . . , Π N, ) and for the disjoint union of its parts.
We call configuration associated to the random point process M N the subset • whose vertices are the l's in C(Π N , M N ), • whose edges connect two indices l and m if d N (Π N,l , Π N,m ) ≤ L, • whose root is the index l = l(x N ) such that δ x N (Π N,l ) = 1, that is to say that x N falls in Π N,l .
We refer to Figure 12 for a drawing of the configuration C(Π N , M N ) and of the two random rooted graphs Γ grid (Π N , M N , L) and Γ geom (X N , x N , d N , M N , L). On this drawing, the space X N is a part of the plane R 2 , the distance comes from the norm · ∞ , the cases of the grid are of size η × η, and L = 3η.  Remark 4.14. In all the proofs hereafter, we shall manipulate atoms of the point processes M N + δ x N or M + δ x , and indices of configurations C(Π N , M N ) or C(Ψ, M); and we shall discuss whether they are connected in a graph Γ geom or Γ grid . When discussing the property of being connected, implicitly, we shall only consider the atoms and the indices that are at graph distance smaller than n from the root of the graph. We ask the reader to keep this convention in mind, which we shall not recall each time and which if omitted might lead to imprecise arguments.
Lemma 4.15. On the event A N,ε , for any L > 0, we have a sequence of inclusions As a consequence of this lemma, if the two discretisations π n (Γ grid (Π N , M N , L − 2η), l(x N )) and π n (Γ grid (Π N , M N , L), l(x N )) are the same and are equal to γ n , then on A N,ε , we also have π n (Γ N , x N ) = γ n . This leads to the inequality for any N ≥ N 0 . From now on, we shall work on X N with the discretised random graphs Γ grid , and the next step of the proof of Theorem 4.11 consists in relating their distribution to events that can be expressed in terms of observables (M N + δ X N )(θ N,l ), where the θ N,l 's are compactly supported continuous functions on X N . To construct these functions, we start from functions compactly supported on B X (x, R + η): Given a configuration C ⊂ [[1, ]], we denote I l (C) = ( 2 3 , +∞) if l ∈ C, and I l (C) = R if l ∈ [[1, ]] \ C. We set The set U C is open in R +1 . On the other hand, for any L > 0, there is a finite set C(γ n , Π N , L) of configurations C ⊂ [[1, ]] such that Γ grid (Π N , M N , L) = γ n if and only if C ∈ C(γ n , Π N , L).
The following lemma relates the event A N,ε ∩ (π n (Γ grid (Π N , M N , L), l(x N )) = γ n ) to the values of the random vector of observables and to its belonging to certain unions of open sets U C .
Lemma 4.16. We then have the following inclusions of events: has at least one atom at distance smaller than η 3 from Ψ l . Since f N,R+η cannot modify the distances by a factor larger than 3 2 , this implies that M N + δ x N has at least one atom at distance strictly smaller than η 2 from Π N,l , and in fact there is exactly one such atom with this property: otherwise, since Π N,l has diameter smaller than η, we would have two distinct atoms at distance smaller than 2η, and this is not allowed on A N,ε . We thus have an injection from C 0 to {atoms of (M N + δ x N ) |Π N }, and this is actually a bijection, because if there were other atoms, then one would have which contradicts the assumption (M N + δ x N )(θ N ) ∈ U C 0 . So, on A N,ε we have a perfect correspondence C 0 ↔ {atoms of (M N + δ x N ) |Π N }. Beware that this does not imply C(Π N , M N ) = C 0 (the two configurations might have occupied cases in the grid Π N that are adjacent but distinct). Let l and m two indices in C 0 , a and b the corresponding atoms, and l and m the indices in C(Π N , M N ) such that a ∈ Π N,l and b ∈ Π N,m . If l is connected to m by γ n , then and as C 0 ∈ C(γ n , Π N , L + η), this implies that l and m are connected by γ n . We conclude that the assumption made at the beginning implies that π n (Γ grid (Π N , M N , L), l(x N )) = γ n , whence the first inclusion of events.
The second inclusion is much simpler. On A N,ε , if π n (Γ grid (Π N , M N , L), l(x N )) = γ n , then C(Π N , M N ) ∈ C(γ n , Π N , L), and By adapting the previous lemma to the events that appear in Inequality (7), we obtain the following: and C∈C(γ n ,Π N ,L−η) and C∈C(γ n ,Π N ,L+η) Since we assume the convergence in the random pointed Lipschitz sense, and since the U C 's are open sets, by the Portmanteau theorem, the right-hand side is larger than the analogue probability involving the limiting point process M + δ x , so and C∈C(γ n ,Π N ,L−η) and C∈C(γ n ,Π N ,L+η) The following final lemma relates the event on the right-hand side to properties of the discretised random geometric graphs on (X, x, d, M): Lemma 4.17. We place ourselves on the event A ε specified before the introduction of the discretised graphs Γ grid . If π n (Γ grid (Ψ, M, L − 4η), l(x)) and π n (Γ grid (Ψ, M, L + 2η), l(x)) are the same graph and are equal to γ n , then (M + δ x )(φ) belongs to C∈C(γ n ,Π N ,L−3η) and C∈C(γ n ,Π N ,L−η) and C∈C(γ n ,Π N ,L+η) U C .
Proof. We suppose that π n (Γ grid (Ψ, M, L − 4η), l(x)) = π n (Γ grid (Ψ, M, L + 2η), l(x)) = γ n , and we are going to prove that C 0 = C(Ψ, M) belongs to C(γ n , Π N , L − 3η), to C(γ n , Π N , L − η) and C(γ n , Π N , L + η). This will imply the result, since by the same argument as in the proof of the previous lemma, (M + δ x )(φ) ∈ U C(Ψ,M) . Let l and m be two indices of C 0 such that Π N,l and Π l,m are at distance smaller than L − 3η. Then, as f N,R+η does not modify the distances by a factor larger than L−2η L−3η , Ψ l and Ψ m are at distance smaller than L − 2η, so l and m are connected in γ n . Conversely, if Ψ l and Ψ m are at distance smaller than L − 4η, then since f −1 N,R+η does not modify the distances by a factor larger than L−3η L−4η , Π N,l and Π N,m are at distance smaller than L − 3η. We conclude that C 0 ∈ C(γ n , Π N , L − 3η), and the two other sets of configurations are treated with similar arguments.
However, we have on A ε the inclusion π n (Γ geom (X, x, d, M, L − 4η), x) ⊂ π n (Γ grid (Π, M, L − 4η), l(x)) ⊂ π n (Γ grid (Π, M, L + 2η), l(x)) ⊂ π n (Γ geom (X, x, d, M, L + 4η), x), for the same reasons as in Lemma 4.15. Since the two bounds given by geometric graphs are the same on A ε and are equal to π n (Γ geom (X, x, d, M, L), x), we have thus shown: As this is true for any ε > 0, and as both sides are probability measures on G • (n), this proves that there is no mass of the distributions of the graphs π n (Γ N , x N ) that escapes at infinity, and that we have in fact lim N→∞ (P[π n (Γ N , x N ) = γ n ]) = P[π n (Γ, x) = γ n ]. This amounts to the local Benjamini-Schramm convergence by Proposition 4.1.
The limiting random graph that appears in Theorem 4.6 is called the (rooted) Poisson Boolean model in [MR96], and it is studied from the point of view of continuous percolation in Chapters 3-5 of loc. cit., as well as in [Pen03, Section 9.6]. The most important result is the existence of a critical parameter λ c (dim G) > 0 for the Poisson point process P = P (λ), such that the resulting random geometric graph with connection distance 1 has no unbounded connected component almost surely if λ < λ c (dim X), and has exactly one unbounded connected component if λ > λ c (dim X); see e.g. [Pen03,Theorem 9.19]. As the random geometric graphs Γ geom (N, L N ) on X converge locally towards the Poisson Boolean model, this implies the following result: 4.4. Convergence in probability of the spectral measures. By Theorem 4.4, the local convergence U(Γ geom (N, L N )) = (Γ N , r N ) → (Γ ∞ , r) implies the weak convergence of the expected spectral measures µ N = E[ν N ] towards a probability measure µ on R. If we want instead to prove the convergence in probability of the spectral measures ν N (without taking the expectation), then taking into account Proposition 4.5, we need to prove the following extension of our Theorem 4.6: Proposition 4.19. Let X be a ssccss, M N the point process on X obtained by taking N independent points v 1 , . . . , v N according to the Haar measure, and r N and r N two independent random vertices in the set of atoms of M N . We denote as before t N = N 1 dim X and d N = t N d, d being the geodesic distance. As N goes to infinity, • the pair of random pointed proper metric spaces ((X, r N , d N , M N ), (X, r N , d N , M N )) converges in the Lipschitz sense towards two independent copies of (R dim X , 0, d Euclidean , δ 0 + P ( 1 vol(X) )); • the pair of random rooted graphs ((Γ N , r N ), (Γ N , r N )) converges in the Benjamini-Schramm sense towards two independent copies of the random graph (Γ ∞ , r) from Theorem 4.6.
Proof. In order to lighten a bit the notations, we shall prove the first item of the proposition when X = G is a Lie group; the proof adapts readily to the non-group case by using the transitive action of the isometry group of X. Fix R > 0, and denote h N,R : B (G,d N ) (e G , R) → B g (0, R) a bijective map which is a quasi-isometry (its dilation constant goes to 1 as N goes to infinity). We then set f N,R (g) = h N,R (g (r N ) −1 ) and f N,R (g) = h N,R (g (r N ) −1 ); these maps are quasi-isometries from B (G,d N ) (r N , R) and B (G,d N ) (r N , R) to B g (0, R). We also consider two continuous and compactly supported functions φ and φ on B g (0, R). We have to show that for any complex numbers z 1 and z 2 ; by replacing φ and φ by z 1 φ and z 2 φ , we can take them equal to 1 in the following. The expectation in Equation (8) is by using the symmetry of the roles played by the variables v 1 , . . . , v N . Here, we convene that φ(h N,R (g)) = 0 if g does not belong to B (G,d N ) (e G , R). • The first term (9) corresponding to the case where r N = r N yields a contribution which is a O( 1 N ), hence negligeable in the limit N → +∞. Indeed, it rewrites as by using on the third line the same arguments as in the proof of Proposition 4.9.
• The second term (10) is asymptotically equivalent to we have only removed the multiplicative factor N−1 N . Setting v = v 1 (v 2 ) −1 , we see that the expectation in Equation (8) is equivalent to If v does not belong to B (G,d N ) (e G , 2R), then the triangular inequality shows that we cannot have at the same time g ∈ B (G,d N ) (e G , R) and gv ∈ B (G,d N ) (e G , R). Therefore, under the condition d N (e G , v) > 2R, we have and the multiplicative factor exp(φ(h N,R (v −1 )) + φ (h N,R (v))) is equal to 1 under this condition. On the other hand, the contribution of the v's such that d N (e G , v) ≤ 2R is asymptotically negligeable, since we are looking at a small ball of volume O( 1 N ). This ends the proof of the first part.
For the second part of the proposition, by using test functions as in the proof of Theorem 4.11, one gets the following easy generalisation of this theorem. Suppose given (X, x 1 , x 2 , d) a bi-pointed proper metric space, and M a random point process on it such that is almost surely continuous at any fixed pair (l 1 , l 2 ). Then, for any pair of positive parameters (L 1 , L 2 ), the map (Γ geom (·, L 1 ) • π 1 , Γ geom (·, L 2 ) • π 2 ) is continuous with respect to the Benjamini-Schramm topology at the point (X, x 1 , x 2 , d, M), where π 1 and π 2 are the two projections on PMS •, of the space PMS ••, of bi-pointed proper spaces endowed with a random point process. The second item of the proposition follows now from the first item, by using the aforementioned generalisation of Theorem 4.11 with the bi-pointed space (X, x 1 , x 2 , d, M) given by two disjoint and independent copies of a Poisson point process on R d .
By applying Proposition 4.5, we finally obtain: Theorem 4.20. Fix a ssccss X and > 0, and consider the random spectral measures ν N of the random geometric graphs Γ geom (N, L N ) on X, with L N = ( /N) 1 dim X . There exists a probability measure µ = µ(dim X, vol(X) ) on R such that we have the weak convergence in probability To close this section, let us propose a slight improvement of this convergence result, which does not seem to be implied by the results from [BL10, BLS11, Bor16] that we presented in Section 4.1.
Proposition 4.21. In the same setting as Theorem 4.20, the measure µ has moments of all order, and it is determined by its moments. We have for any s ≥ 1 where the convergence occurs in L 2 (and therefore also in probability).
Proof. We start by examining the moments M s,N = R x s µ N (dx) of the expected spectral measures µ N . By design, the adjacency matrices of our graphs have zeroes on their diagonal, so M 1,N = M 1,N = 0. Suppose now that s ≥ 2. We can rewrite M s,N as follows: and we have the following inequalities: ∑ 1≤i 1 ,...,i s ≤N no consecutive indices are equal , then M s,N ≤ M s,N , and by using the independence of the vectors v i and the inequality vol X (B(v i , L)) ≤ vol R dim X (B(0, L)) which holds since a sscc symmetric space has (constant) positive curvature, we obtain: By induction and since M 2,N ≤ c(dim X) vol(X) , we conclude that for any s ≥ 2, By Fatou's lemma, since µ N µ, the same upper bound holds for the even moments of µ: where the implied constant in the O(·) only depends on the space X and . By Carleman's criterion (see for instance [Bil95, Chapter 30]), µ is therefore determined by its moments. where (Γ ∞ , r) is the rooted Poisson Boolean model appearing as the limit in Theorem 4.6. The Benjamini-Schramm convergence ensures that each term of the series for M s,N converges towards the corresponding term for M s , therefore, to prove that M s,N → M s , we only need a uniform domination on the terms of these series M s,N . If γ s is an element of G • (s) with k + 1 vertices, then the number of rooted s-cycles in γ s is smaller than k s−1 . On the other hand, the probability that π s (Γ N , r N ) has k + 1 vertices is smaller than the probability that at least k vertices v i fall in B X (o, s L N ), that is Remark 4.22. Standard arguments from the theory of convergence of measures show that the convergence in probability of all the moments M s,N → M s is stronger than the convergence in probability ν N µ. In particular, the proof above can be used to bypass the general arguments from Section 4.1 that connect the local convergence of graphs to the weak convergence of their spectral measures.

CHARACTERS
In this last section before the appendices, we focus on the case of a sscc Lie group G, and we investigate the connections between: • its representation theory and the formulas obtained in Section 3 for the asymptotics of the Gaussian regime; • the limiting measure µ = µ(dim G, vol(G) ) exhibited in Section 4 and that drives the asymptotics of the Poissonian regime.
An important objective is to obtain more information on the limiting measure µ, thereby answering the following questions: • Is µ compactly supported? What is the growth rate of the moments M s of the measure µ?
• Does the measure µ admit atoms, or is it absolutely continuous with respect to the Lebesgue measure?
In the proof of Proposition 4.21, one can try to make Equation (11) more precise, and to gather the cycles that one needs to count according to the identities of indices i 1 , . . . , i s that might occur. This theory leads to a combinatorial circuit expansion of the moments M s,N and of their limit M s , which we develop in Section 5.1. This combinatorial expansion of the moments involves directed graphs endowed with a distinguished traversal, and these circuits can be reduced to yield non-directed graphs possibly with loops and with labels on their edges. For instance, we shall prove that we have an expansion , each term of this expansion being a monomial in the parameter , and corresponding to the limit of a certain observable of the random geometric graph Γ geom (N, L N ). As explained in the introduction, the actual computation of each of these terms should be performed by using the flat model T dim G , where representation theory is encoded by classical Fourier series. However, if one stays with the non-flat sscc Lie group G, then the same computations shed light on several important phenomena from asymptotic representation theory, and this approach allows one to understand clearly the degeneration from the Gaussian to the Poissonian regime (see Sections 5.2 and 5.3). Even more importantly, it leads to Conjecture 1.4, which we detail in Section 5.4. Although we do not see yet how to solve it, we consider it to be one of the main result of our study, which is why we devoted this section to its presentation.
Before presenting the combinatorial expansion of the moments M s , we should warn the reader of two things: (1) We do not plan to compute here explicitly all the moments M s (or at least to obtain some precise upper bounds on them). The arguments of Section 5.1 and the replacement of the space G by its flat model mostly reduce these calculations to a combinatorial problem of counting graphs with certain weights, but even with these reductions, these enumerations are by no means easy to perform. We hope to address this problem in a forthcoming work.
(2) Secondly, the phenomena from asymptotic representation theory in Sections 5.2-5.4 quickly rely on certain algebraic arguments which are more advanced than before, namely, the theory of crystals and string polytopes of Lusztig-Kashiwara and Berenstein-Zelevinsky. The precise form of our Conjecture 1.4 also relies on this theory. In order to ease the reading of this section, we shall try to present our arguments without insisting too much on these algebraic prerequisites; they can be found in an appendix at the end of this article (Section 7), which is a short survey of some results regarding the crystals of representations of Lie groups.

Circuit expansion of the expected moments.
Until the end of Section 5, G is a connected compact Lie group endowed with a bi-invariant Riemannian structure, and starting from Subsection 5.2 we shall assume it to be simple and simply connected. We consider the Poissonian geometric graph on G with parameters N and L N given by Equation (4); in particular, the parameter is fixed from now on, and most of the quantities manipulated hereafter implicitly depend on it (for instance, the expectations E H,T,N defined below). In this paragraph, we give a combinatorial expansion of M s,N = E[ν N (x s )] = µ N (x s ) and of M s = lim N→∞ M s,N in terms of circuits; this is the first step towards the calculation of the moments M s of the limiting measure µ. We assume s ≥ 2 since M 1, where the v i 's are independent Haar distributed random variables on G, and the sums run over indices i j ∈ [[1, N]] such that two consecutive indices i j and i j+1 are never equal (by convention, the index following i s is i 1 ). Now, an expectation E i 1 ,i 2 ,..., only depends on the possible equalities of indices. For instance, when computing M 4,N , we have: The first term corresponds to the case where all the indices i 1 , i 2 , i 3 , i 4 are distinct; the second term corresponds to the identities i 1 = i 3 or i 2 = i 4 ; and the last term is when i 1 = i 3 and i 2 = i 4 simultaneously. We associate to these four cases the circuits of Figure 13. By circuit, we mean a directed graph H, possibly with multiple edges but without loops, endowed with a distinguished traversal T that goes through each directed edge exactly once, and that is cyclic (the starting point is the same as the end point of the traversal). We identify two circuits (H 1 , T 1 ) and (H 2 , T 2 ) if there exists a graph isomorphism ψ : C 1 → C 2 that is compatible with the traversals, that is ψ(T 1 ) = T 2 . Given a circuit (H, T) with s edges and k ≤ s vertices, we associate to it the expectation of a function of k independent points v 1 , . . . , v k on G: Notice that E H,T,N only depends on H, and not on the particular traversal T, because each directed edge of H appears exactly once in T. One can recover the identities from the corresponding circuit, and any circuit of length s corresponds to a set of identities of indices, without identities i j = i j+1 since we do not allow loops. Moreover, the number of terms in the sum M s,N corresponding to a circuit with k vertices is N(N − 1) · · · (N − k + 1); and each term corresponding to a circuit (H, T) is equal to 1 N E H,T,N . This ends the proof of the expansion of M s,N over circuits.
The calculation of the quantities E H,T,N involves the operation of reduction of circuit. Let (H, T) be a circuit of length s. Its reduction is the labeled undirected graph which is allowed to be disconnected and to have loops, and which is obtained by performing the following operations: • forgetting the orientation of the edges of H; • replacing any multiple edge by a single edge; • putting a label 1 on each of the (single) edges; • cutting the graph at each of its cut vertices (also called articulation points), replacing a configuration L 1 L 2 with L 1 = ∅ and L 2 = ∅ by • finally, replacing the connected components 1 by loops 2 .
The fourth operation in the algorithm of reduction splits the graph in its so-called biconnected components: they are connected components which remain connected if one removes one vertex. Notice that the operation of reduction: • can send many distinct circuits to the same reduction; • can create two kinds of connected components: labeled loops based at one single vertex and with a label greater than 2; and connected loopless graphs on at least two vertices, all of them being at least of degree 3.
Example 5.2. The reduction of the circuit of Figure 14 appears in Figure 15. Similarly, the reductions of the four circuits of length 4 are drawn in Figure 16, with the middle one that has multiplicity 2 (as well as two connected components). If R is the reduction of a circuit (H, T) with |H| = k vertices, then one has the identity: where l e is the label of an edge e in R; k is the number of vertices of R; and c is the number of connected components of R. Equation (12) shows readily that R → k − 1 is an additive map with respect to connected components of reduced circuits.

Lemma 5.3. The expectation E H,T,N only depends on the reduction of the circuit (H, T).
Proof. Consider the expectation E H,T, • Since h N is a symmetric kernel, it does not depend on the orientation of the edges in H; this allows the first step in the reduction of the circuit. N (x, y)) for any m ≥ 1, so one can replace multiple edges by simple edges in the graph.
• The factorisation of E H,T,N on the biconnected components of the graph is a consequence of the independence of the vertices v i , and of the invariance of the function h N by action of the group G on its two variables. Indeed, suppose that the labeled graph L obtained after the three first steps of the reduction has two components L 1 and L 2 which only share one cut vertex v c . Then, there are two disjoint sets of vertices {v i 1 , . . . , v i r } and {v j 1 , . . . , v j s } and two functions f 1 and f 2 such that Moreover, the two functions f 1 and f 2 are products of functions h N (x, y), with x and y in the set of variables of f 1 or f 2 . This implies that for any g ∈ G, f 2 (v c , v j 1 , . . . , v j s ) = f 2 (gv c , gv j 1 , . . . , gv j s ). We take g uniform under the Haar measure, and we set v c = gv c and v j k = gv j k . Then, with two set of variables that are now independent.
• Take now a connected component after the factorisation in biconnected components. If one has in such a graph a sequence of r edges · · · · · · , then it corresponds to a product in the expectation, with the independent random variables v b 1 , . . . , v b r−1 that do not appear anywhere else in the product in the expectation E H,T,N ; clearly one can encode this term by a labeled edge r . This is equivalent to the second last rule of reduction.
• The last step is a convention that will allow us to have only two kinds of connected components, namely, the labeled loops, and the labeled loopless graphs on at least two vertices which all have degree larger than 3. It amounts to the obvious identity In the following, we denote R(H, T) the reduction of a circuit (H, T). Note that R(H, T) depends only on H (and even the underlying undirected graph). However, since we shall consider sums over circuits, it is more convenient to recall each time the pair (H, T). The previous discussion leads to: Theorem 5.4 (Circuit expansion).
(1) Combinatorial expansion: for any s ≥ 0, we have where the sum runs over the finite set of circuits with s edges, and where E R(H,T),N = E H,T,N only depends on the circuit reduction of (H, T).
(2) Factorisation: if R = R 1 R 2 · · · R c , then E R, (3) Asymptotics: for any reduced circuit R with parameter k given by Equation (12), there exists a positive real number e R depending only on dim G such that Therefore, Proof. The first part of the theorem comes from the combination of Lemmas 5.1 and 5.3, and the second part corresponds to the fourth step of the algorithm of reduction of circuits. It remains to examine the asymptotics of E R,N as N goes to infinity. We denote N ↓k = N(N − 1) · · · (N − k + 1) a falling factorial. Given a reduced circuit R = R(H, T) and a rooted graph γ ∈ G • , we call embedding of the circuit (H, T) into γ an injective morphism of graphs e : V(H) → V(γ) which sends the starting and ending point of the traversal T to the root of γ. We then have: the limit that we have obtained; the last thing that remains to be shown is that e ( ) (H,T) depends polynomially on the intensity vol(G) of the Poisson point process underlying (Γ ∞ , r). However, the expected number of embeddings can be rewritten as the integral of the (k − 1)th factorial moment measure M (k−1) of the Poisson point process P ( volG ) against a certain Borel measurable subset in (R dim G ) k−1 : where we convene that the vertices in H are labelled by the integers in [[0, k − 1]], and that x 0 = 0 in R dim G . We refer to [DVJ03, Chapter 5] for details on the notion of factorial moment measure; it is well known that for the Poisson point process with intensity µ, the r-th factorial moment measure is simply µ ⊗r . This proves the dependence stated in the theorem, since we have here .

Asymptotic contribution of a reduced circuit which is a loop. The remainder of this section is devoted to the study of the connection between:
• the coefficients e ( ) (H,T) , which we sometimes also index by the corresponding reduced circuits R = R(H, T) and denote e ( ) R ; • the representation theory of the group G, which from now on will be assumed to be sscc.
Although we know that e ( ) (1) R k−1 , in the following it will be convenient to keep the coefficient : it will enable one to keep track of the dimensions of various rescalings that we shall perform. The existence of the limits e ( ) R is strongly related to some interesting results or conjectures in asymptotic representation theory. To understand how the representation theory of G drives the degeneration from the Gaussian to the Poissonian regime, we start by examining the case of a circuit (H, T) which is a simple cycle of length k ≥ 2, and thus has reduction x), then by using the invariance of distances by the action of G, we can rewrite where the scalar product is taken in the convolution algebra L 2 (G, dg) (and even in the subalgebra L 2 (G) G ). In this Hilbert space, we have the decompositions ch λ for any irreducible representations λ, µ ∈ G. Therefore, where d = rank(G) denotes as in Section 3 the rank of G. We put an index N on C λ,N to insist on the dependence on N of the Fourier coefficients of Z L N . We have thus shown: As N goes to infinity, this series will transform into a Riemann sum and converge towards an integral involving Bessel functions. Let us start by evaluating the asymptotics of C λ,N when N grows and the parameter x = L N (λ + ρ) is fixed in the Weyl chamber C. We shall use the following properties of the function J RΩ : • it is a smooth function on RΩ with maximum value • it is invariant by rotations; • its asymptotics are (see [Coh07, Proposition 9.8.7]) In particular, any power k ≥ 2 of the function J RΩ is integrable on RΩ. For x ∈ C, set On the other hand, for any (positive) root α and any smooth function f on RΩ, we define the partial derivative where K RΩ (L N , x) is a function on the translated Weyl chamber C + L N |Φ + | ρ such that, uniformly for L N small enough, Proof. Recall that in the group algebra of the space of weights RΩ, we have the identity see [Bum13,Proposition 22.7]. Therefore, if an element ω ∈ RΩ acts on smooth functions f by the operator (e ω f )(x) = f (x − ω), then we can write: For k ≥ 0, let F (C + kρ) be the set of smooth functions f on the translated Weyl chamber C + kρ, such that any partial derivative ((∏ r i=1 ∂ δ i ) f )(x) is bounded by K(δ 1 , . . . , δ r ) min 1, 1 x rank(G)+1 2 . We claim that: (1) The modified Bessel function J RΩ belongs to the class F (C).
(2) If f belongs to F (C + kρ), then for any positive root α, and any x ∈ C + (k + 1)ρ e L N α where g(L N , x) belongs to F (C + (k + 1)ρ), and where the bounds on the partial derivatives ((∏ r i=1 ∂ δ i )g(L N , ·)) are uniform in L N (for L N small enough). The first claim follows from the asymptotic estimate of Bessel functions and from the recurrence relation ( J β ) (x) = −x J β+1 (x). The second claim is obtained by a Taylor expansion of the function f around x. To ensure that one can use it, one needs to translate x a bit further inside the Weyl chamber, which is why the estimate holds only in C + (k + 1)ρ if f ∈ F (C + kρ). By combining the two claims and taking |Φ + | discrete derivatives of J RΩ , one gets the result of the lemma.
We set . Therefore, by using also the relation d + 2 |Φ + | = dim G which follows from the decomposition of the adjoint representation of G in root subspaces, we obtain for any x = L N (λ + ρ) falling into the translated Weyl chamber C + L N |φ + | ρ. Therefore, with a remainder that is a O(L N ), because it consists of: • the contribution of the weights λ that are in the boundary C \ (C + |Φ + |ρ) of the Weyl chamber; • and terms proportional to We leave the reader to check that these contributions can indeed be summed and yield a O(L N ); this relies on estimates of Bessel functions similar to those previously given. Then, we are left with a standard Riemann sum over the lattice L N (C ∩ ZΩ), whose points correspond to domains of volume (L N ) d vol(RΩ/ZΩ). We have a duality of lattices and vol(RΩ/ZΩ) = 1 vol(t/t Z ) .
We conclude: Theorem 5.6. For any k ≥ 2, we have where dx is the Lebesgue measure on RΩ associated to the scalar product of weights defined in Section 2.3; d = rank(G); and δ( More precisely, the difference between N k−1 E H,T,N and its limit is a O(L N ), with a constant in the O(·) that only depends on G and s. In the following, since we shall always deal with the partial derivative (−1) |Φ + | ∂ Φ + = ∂ Φ − = ∏ α∈Φ − ∂ α , it will be convenient to use the latter notation ∂ Φ − . As an application of Theorem 5.6, we can compute the limits M s = lim N→∞ E[ν N (x s )] for any s ∈ {2, 3, 4, 5}. Indeed, we can enumerate all the circuits of length s ≤ 5, and all their reductions have connected components which are loops: . Therefore, given a sscc Lie group G, if we set for k ≥ 2 then, the five first asymptotic moments of the spectral measure of Γ geom (N, L N ) with L N = ( /N) 1 dim G are given by: M 4 = I 4 ( ) 3 + 2 (I 2 ) 2 ( ) 2 + I 2 ; M 5 = I 5 ( ) 4 + 5 I 3 I 2 ( ) 3 + 5 I 3 ( ) 2 , where = vol(t/t Z ) .

Asymptotic contribution of a connected reduced circuit with two vertices.
What is important in the previous paragraph is not the explicit formula that one obtains for the five first moments, but the method that leads to it: indeed, if one tries to extend it to higher moments, then one is led to new results in representation theory. These results and conjectures are related to the theory of crystals, and in order to understand this, one can try to compute the sixth moment of µ with the same method as above. The the enumeration of all the circuits of length 6 yields The asymptotics of E R,N with R as in Figure 17 are related to the asymptotics of tensor products V λ ⊗ V µ when x = L N λ and y = L N µ are fixed points in the interior of the Weyl chamber, and L N goes to 0; thus, λ and µ are very large dominant weights. Indeed, we have The functions above decompose in L 2 (G) as: Therefore, we have As before, the idea is to consider the sum above as a Riemann sum, and we will of course use Lemma 5.5 in order to approximate the coefficients C λ,N , C µ,N and C ν,N by partial derivatives of Bessel functions. However, we also need to deal with c λ,µ ν = ch λ × ch µ ch ν , and the product of characters ch λ × ch µ is the character of the tensor product of representations V λ ⊗ V µ . Therefore, we need to understand the asymptotics of the Littlewood-Richardson coefficients c when λ and µ are very large. More generally, if we are interested in the computations of the terms e R where R is a general reduced circuit on two vertices, then we need to understand the asymptotic behavior of the Littlewood-Richardson coefficients of tensor products with more than 2 irreducible representations. Given dominant weights λ 1 , . . . , λ r−1≥2 , we write These generalised Littlewood-Richardson coefficients are connected to the usual one by the convolution rule: c Proposition 5.7. We denote d the rank of the sscc Lie group G, and l = |Φ + | its number of positive roots. We fix directions x 1 , . . . , x r−1≥2 in the interior C of the Weyl chamber. There exists a compactly supported piecewise polynomial function q x 1 ,...,x r−1 (z) on C such that: • This function is non-negative and symmetric in x 1 , . . . , x r−1 .
• For any bounded continuous function f on C, • The function q x 1 ,...,x r−1 (z) is related to the functions q x,y (z) by the convolution rule: • The function (x 1 , . . . , x r−1 , z) → q x 1 ,...,x r−1 (z) is locally a homogeneous polynomial function of total degree (r − 2)l − d. The domains of polynomiality of this function are polyhedral cones in C r .
• The total mass of the positive measure q x 1 ,...,x r−1 (z) dz is smaller than This proposition is proved in the second appendix of this paper; the proof relies deeply on the theory of crystal bases and string polytopes. Let us give an intuitive explanation of it which does not go too much into the algebraic details. Given three dominant weights λ, µ and ν, one can construct a polytope P(λ, µ) (the relative Berenstein-Zelevinsky polytope) in a vector space of dimension l = |Φ + | which is determined by hyperplanes whose equations depend on λ and µ in an affine way. Then, the Littlewood-Richardson coefficient c λ,µ ν is the number of integer points which lie in the intersection of this polytope and of a vector subspace determined by d = rank(G) equations, these equations depending on λ and ν again in an affine way (see Theorem 7.8). When we consider a sum ∑ ν∈ G c tx,ty ν f ν t with x, y ∈ C , tx, ty ∈ G and t going to infinity, the polytope P(tx, ty) scales linearly with the parameter t, and the counting measure of the integer points of this polytope becomes after scaling the uniform Lebesgue measure on P(x, y), which is of dimension l. The Riemann sum over the dominant weights ν becomes after scaling an integral against the affine projection of this uniform measure on a polytope of smaller dimension d. Since an affine projection of a uniform measure on a polytope is a piecewise polynomial function also supported by a polytope, this essentially proves Proposition 5.7 in the case r = 3, by using also the linearity of the various polytopes in the parameters λ, µ, ν. The general case r ≥ 3 follows by using the convolution rule for multiple Littlewood-Richardson coefficients. In the sequel of this section, we use Proposition 5.7 without insisting on its algebraic origin. The knowledge of these algebraic beginnings will only be useful in order to understand fully our conjecture on graph functionals, and why it involves the enumeration of integer points in polytopes; again, we refer to Section 7 for more details.
We now come back to the asymptotics of E R,N when R is a connected reduced circuit with two vertices. An arbitrary connected reduced circuit on k = 2 vertices writes as a 1 a 2 a 3 · · · a r R = .
with a 1 ≥ a 2 ≥ · · · ≥ a r ≥ 1, at most one index a r = 1, and r larger than 3. In this setting, k = (a 1 + · · · + a r ) − (r − 2). The contribution corresponding to such a reduced circuit is: the sum running over r-tuples of dominant weights. With t N = (L N ) −1 , by combining Proposition 5.7 and the estimates of the coefficients C λ,N given by Lemma 5.5, we obtain: the sums running over elements x 1 , . . . , x r which are in the Weyl chamber C, and which are multiple by L N of some dominant weights. The convergence of the integral on C r follows from the following argument. By using the bounds on the partial derivatives of J RΩ , and the scaling properties of the functions δ(x i ) and q x 1 ,...,x r−1 (x r ), one sees that it suffices to prove the convergence at infinity of the integral ∞ t (r−2)l−d−(a 1 +···+a r ) d+1 2 −(a 1 +···+a r −r)l t r(d+1)−1 dt. Indeed, q x 1 ,...,x r−1 (x r ) is homogeneous with total degree (r − 2)l − d; we have the upper bound and ∏ r i=1 (δ(x i )) a i −1 is homogeneous with total degree (a 1 + · · · + a r − r)l. Therefore, we have to prove that However, the worst case is when the a i 's are minimal, that is a 1 = a 2 = · · · = a r−1 = 2 and a r = 1. The left-hand side of the inequality above is then equal to which is clearly negative. On the other hand, the validity of the approximation of the Riemann sum by an integral follows from the smoothness of the functions considered, and from the fact that the functions q x 1 ,...,x r−1 (x r ) are compactly supported. We have therefore proved: Theorem 5.8. Let r ≥ 3, a 1 ≥ a 2 ≥ · · · ≥ a r−1 ≥ 2 and a r ∈ [[1, a r−1 ]]. We set e (l) a 1 a 2 · · · a r , N where (H, T) is a circuit whose reduction is a connected component on two vertices with parameters (a 1 , . . . , a r ). Here, k = (a 1 + · · · + a r ) − (r − 2). Then, the function q x 1 ,...,x r−1 (x r ) being related to the asymptotics of (multi-)Littlewood-Richardson coefficients by Proposition 5.7.
As an application, if we set then this quantity is related to the term e ( ) R with R as in Figure 17, and by using the circuit expansion of M 6 previous computed, we obtain M 6 = I 6 ( ) 5 + (6 I 4 I 2 + 3 (I 3 ) 2 ) ( ) 4 + (6 I 4 + 6 (I 2 ) 3 + 9 I (2,2,1) ) ( ) 3 + (6 (I 2 ) 2 + 4 I 3 ) ( ) 2 + I 2 , where = vol(t/t Z ) . Again, the important point is not this exact formula, but the fact that its computation sheds light on the asymptotic properties of large representations of the group G. This idea culminates in the computation of the higher moments M s≥8 , as we shall now explain -for s = 7, one can check that the circuit expansion is where E(R) is the set of labeled edges of R. Note that the integral on the second line only depends on the unlabeled oriented graph S underlying the reduced connected circuit R. In the following we always use the letter S or the notation S(H, T) for this unlabeled oriented graph, and we call graph functional the integral this is a function on the k -tuples of dominant weights indexed by the edges of S. This definition actually makes sense for any finite graph S with ordered vertices and possibly with loops; moreover, the definition immediately implies that GF S ((λ e ) e∈E(S) ) factorises over the biconnected components of the graph S. We recover classical quantities when the graph S has k = 1 or k = 2 vertices: • If the graph S has k = 1 vertex, then it is a collection of loops, and we have • If the graph S has k = 2 vertices and r ≥ 3 edges between them, then its graph functional is a multiple Littlewood-Richardson coefficients: where λ * denotes the highest weight of the irreducible representation which is the conjugate of V λ (i.e., (V λ ) * = V λ * and ρ λ * (g) = (ρ λ (g −1 )) t ). Hence Now, let us do a little bit of dimension analysis in order to explain how the mere existence of the limits e ( ) R suggests our Conjecture 1.4. To simplify a bit the discussion, we suppose in the sequel that R(H, T) and S(H, T) are connected. The idea is to interpret Equation (14) as a Riemann sum, which should be asymptotically of order O(N −(k−1) ), k being as usual the number of distinct indices in the circuit (H, T). If x e = λ e L N , then we should be able to approximate where r = |E(R)| and k = |V(R)|. Assuming that this approximation is valid, we then have: , and where the sum runs over elements of the lattice (L N (C ∩ ZΩ)) r . If we want the convergence of this Riemann sum as N goes to infinity, then taking into account our Proposition 5.7 and our Theorems 5.6 and 5.8, it is natural to make the following assumption: If r is the number of edges of S and k, k are as in Equation (12), then there exists a function q S : C r → R + which is locally polynomial, with domains of polynomiality that are polyhedral cones and with total degree lr − (2l + d)(k − 1) in these cones, such that we have the asymptotics if the tx e 's are dominant weights in the interior of the Weyl chamber.
Actually, this estimate cannot be true for any family of dominant weights (tx e ) e∈E(S) , because a graph functional GF S ((λ e ) e∈E(S) ) usually vanishes outside a full rank sublattice of (ZΩ) r . For instance, with the Littlewood-Richardson coefficients, c λ,µ ν = 0 if λ + µ − ν does not belong to the root lattice, which is smaller than the weight lattice. Thus, the correct estimate should rather be GF S ((tx e ) e∈E(S) ) t→∞ t lr−(2l+d)(k −1) q S ((x e ) e∈E(S) ) if (tx e ) e∈E(S) ∈ A S , 0 otherwise, where A S ⊂ (ZΩ) r is a sublattice with maximal rank rd. Assuming that this is true, we would then obtain the analogue of Theorems 5.6 and 5.8 for any connected reduced circuit R(H, T), with l e 2 (δ(x e )) l e −1   q S (x 1 , . . . , x r ) dx 1 · · · dx r .
Finally, our assumption on the asymptotic behavior of the graph functionals would follow immediately from the fact that these functionals GF S count the integer points in certain polytopes P((λ e ) e∈E(S) ) whose equations are determined by affine functions of the dominant weights; as explained just after the statement of Proposition 5.7 and in more details in Section 7, this is the case when S consists of two vertices. We now have fully explained our Conjecture 1.4, except for the belonging of the polytopes to the so-called string cone of the group G r ; this is also expained in the appendix at the end of the paper. We hope to be able to prove the conjecture by interpreting the graph functionals in the theory of crystal bases, and by describing them in terms of string parametrisations. We close this section by two remarks.
Remark 5.9. The formula lr − (2l + d)(k − 1) does not give a non-negative number for any connected graph S; for instance, if G = SU(2) and S is the connected oriented graph associated to the reduced circuit from Figure 18, then l = d = 1, r = 5 and k = 3, therefore lr − (2l + d)(k − 1) = 5 − 6 = −1. In this situation, the corresponding polytope should be empty, and our conjecture should imply some vanishing results, which can be stated informally as follows: if one takes a graph functional of irreducible representations with too many Haar distributed random variables g 1 , . . . , g k in comparison to the number r of characters appearing, then this integral vanishes. This is not very surprising since G ch λ (g) dg = 0 for any non trivial representation, but our conjecture would make this much more precise.
Remark 5.10. In the general case, it is certainly hopeless to have a beautiful closed formula for the functions q S introduced above. It should however be noticed that the analogue of Conjecture 1.4 is trivially true when G is a torus T d (this is not a semisimple Lie group, but the whole theory adapts mutatis mutandis). In this case, the weights of irreducible representations are elements of Z d , and one can check that the graph functionals are indicator functions of sublattices of (Z d ) r . This means that M s can always be written as a sum over reduced circuits of certain weights which are integrals of combinations of Bessel functions, but this time without a complicated locally polynomial function as the measure of integration. This seems a promising approach for the problem of computing a precise upper bound on M s ; note however that we need a good control of these weights if we want to improve substantially the arguments from Proposition 4.21.

APPENDIX: GEOMETRY OF THE CLASSICAL SSCC LIE GROUPS
In this appendix, we describe for each classical case: the maximal torus T; the Weyl group W; the corresponding weight lattice ZΩ and root system Φ; the set of dominant weights G; the dimensions of the corresponding irreducible representations. We also compute the volumes of the classical sscc Lie groups with respect to the Riemannian structure given by Equation (1) and the opposite of the Killing form. Most of these results can be found in the classical text books [Hel78,FH91], and we stick to the conventions of a previous paper [Mél14]. Since Spin(n) is not easily described in terms of matrices, in the following we shall use numerous times the two-fold covering map π : Spin(n) → SO(n).
6.2. Weyl groups. The Weyl groups W = Norm(T)/T corresponding to the previous choices of maximal tori are: type A n : W = S(n + 1), acting by permutation of the angles; type B n : W = (Z/2Z) S(n), acting by permutation and inversion of the angles; type C n : W = (Z/2Z) S(n), acting by permutation and inversion of the angles; type D n : W = ((Z/2Z) S(n)) even , acting by permutation and inversion of the angles.
The dominant weights in G are the positive integer combinations of these fundamental weights. They have positive scalar products with the positive roots: We have drawn in Figure 19 the weight lattices, the root systems and the Weyl chambers in rank 2.
. The weight lattices in type A 2 , B 2 , C 2 and D 2 .
For each (simple) root α, there is a unique vector T α ∈ t C such that α(t) = B(t, T α ). The scalar product on RΩ is then given by α | β = B(T α , T β ). One thus obtains the following scalar products: • Each time, the vectors e i form an orthogonal basis, with the following square norms: • In type A n , the weight space RΩ is embedded in Span R (e 1 , . . . , e n+1 ) as the hyperplane RΩ = Span R (α 1 , . . . , α n ) with α i = e i − e i+1 (the α i 's are the simple roots).
• In type B n , C n and D n , the weight space RΩ is Span R (e 1 , . . . , e n ).
Note that in many representation theoretic formulas, one does not need to know exactly the normalisation of the vectors e i , because one deals with quotients of scalar products (for instance, in Weyl's dimension formula). However, the knowledge of the normalisation is required for instance in Theorem 3.1, and in several other theorems stated in this paper.

APPENDIX: CRYSTALS OF REPRESENTATIONS AND STRING POLYTOPES
This second appendix proves Proposition 5.7 and gives a survey of the theory of crystals of representations. We have tried to explain it in the most pedagogical way that we were able to, and in particular we start with the path model, although it is not really required in our study. Until the end, G is a fixed sscc Lie group, g is its Lie algebra, and d is the rank of G. The set of simple roots of G is denoted (α i ) i∈ [[1,d]] . This is a linear basis of RΩ, and we denote (α ∨ i ) i∈[[1,d]] the basis of simple coroots in (RΩ) * , defined by the relations consists in the fundamental weights, such that ZΩ = Span Z (ω 1 , . . . , ω d ). Fix a dominant weight λ ∈ G, and for ω ∈ ZΩ, denote V λ (ω) the weight subspace of V λ associated to the weight ω: An element of the weight subspace V λ (ω) is called a weight vector of V λ , and the irreducible representation V λ is the direct sum of its weight subspaces: The set of weights with positive multiplicity in V λ will be denoted Ω(λ); it is a finite subset of λ + R, where R is the root lattice of G, that is the sublattice of ZΩ spanned by the (simple) roots.
7.1. Crystals and the path model. The theory of crystal bases and the path model allow one to compute the multiplicities K λ,ω = dim C (V λ (ω)) for ω ∈ Ω(λ) (they are also called the Kostka numbers). Let U q (g C ) be the quantum group of the complexification g C of the Lie algebra g; it is a deformation with a complex parameter q of the universal enveloping algebra U(g C ), see [Jim85,Jim86]. There is a corresponding deformation V λ q of the irreducible module V λ , and a notion of weight vectors in V λ q , such that if then the weights and the multiplicities are the same for V λ and for V λ q : This is the Lusztig-Rosso correspondence, see the original papers [Lus88,Ros88,Ros90], and [Mél17, Chapter 5] for a detailed exposition of the case g = gl(n). The correspondence holds for any q which is not 0 or a root of unity. Now, a crystal basis of the irreducible representation V λ q is a linear basis C(λ) of V λ q that consists of weight vectors, and such that if (e i , f i , q h i ) i∈ [[1,d]] are the Chevalley generators of U q (g C ), then for any vector v of the crystal basis, e i · v is either 0 or another vector v of the crystal basis; and similarly for f i · v. Notice that if v ∈ C(λ) has weight ω and v = e i · v (respectively, v = f i · v) does not vanish, then v has weight ω + α i (respectively, ω − α i ). The crystal of V λ q is given by a crystal basis C(λ), and by the weighted labeled oriented graph: • with vertices v ∈ C(λ), • with a weight map wt(·) which associates to v ∈ C(λ) the corresponding weight in ZΩ.
It has been shown independently by Lusztig and Kashiwara that crystal bases of irreducible representations of semisimple Lie algebras always exist, and that their combinatorial structure does not depend on q; see [Lus90,Kas90]. In the sequel, we shall only work with the combinatorial object (weighted labeled oriented graph). Indeed, if one knows the crystal of an irreducible representation V λ , then one recovers immediately the highest weight of the representation, and all the multiplicities of the weights: for any ω ∈ Ω(λ), dim C V λ (ω) = card{v ∈ C(λ) | v has weight ω}.
As a consequence, we can now forget the underlying quantum groups U q (g C ).
This decomposition is better understood in a picture, see Figure 20 for an example on the weight lattice of type A 2 . Denote s α the reflection with respect to the root α, that is the map x → x − 2 x | α α | α α. For j ∈ [[1, ]], we define π j = s α (π j ) if π j is of type (1), π j if π j is of type (2).
The crystal C(π λ ) that one obtains is finite, and it is isomorphic to the crystal C(λ) of the irreducible representation V λ . In particular, dim C V λ (ω) = card{paths in the crystal C(π λ ) with endpoint ω}, and the character ch λ is given by the formula ch λ = ∑ π∈C(π λ ) e wt(π) .
Actually, one can take instead of π λ any path from 0 to λ that stays in the Weyl chamber; all these paths generate the same crystal C(λ).
In particular, each of the six roots of SU(3) has multiplicity 1 in the adjoint representation, whereas the weight 0 has multiplicity 2 = rank(SU(3)).
Let us now explain the use of the path model in order to compute tensor products. If V λ ⊗ V µ = ∑ ν∈ G c λ,µ ν V ν , then the concatenation product of crystals C(π λ ) * C(π µ ) is a set of paths such that the action of the root operators on these paths generate a crystal whose connected components are isomorphic to the elements of the multiset {(C(π ν )) m ν , ν ∈ G}. Therefore, c λ,µ ν is equal to the number of paths π ∈ C(π µ ) such that π λ * π always stays in the Weyl chamber C, and π λ * π ends at the dominant weight ν; see [Lit98b, Proposition 2 and Corollary 1]. In a moment, we shall reinterpret this rule in the string polytope of V µ , see Theorem 7.8.
Remark 7.3. This link between the tensor product of representations and the concatenation product of crystals proves that the set of dominant weights ν such that c λ,µ ν > 0 is included in R + λ + µ, where R is the root lattice. Indeed, the crystal C(π µ ) consists of paths with weights in µ + R, and if c λ,µ ν > 0, then there is a path in this crystal that connects 0 to ν − λ.
7.2. The cone and the polytopes of string parametrisations. In this paragraph, we fix a dominant weight λ, and a decomposition of the longest element w 0 of the Weyl group as a product of reflections s α i along the walls of the Weyl chamber C: w 0 = s α i 1 s α i 2 · · · s α i l .
Notice that l = |Φ + | is equal to the number of positive roots of G. If v is an element of the crystal C(λ), we call string parametrisation of v the vector of integers (n 1 , n 2 , . . . , n l ) ∈ N r such that: • n 1 is the maximal integer such that e n 1 α i 1 (v) = 0; • if n 1 , . . . , n s−1 are known, then n s is the maximal integer such that e n s α i s · · · e n 2 α i 2 e n 1 α i 1 (v) = 0.
An explicit description of the string cone is given in [Lit98a,BZ01]; see also the remark after Theorem 7.8. On the other hand, the string polytope P(λ) has maximal dimension l as long as λ does not belong to the walls of the Weyl chamber.
This polytope is drawn in Figure 24, and one can check that it contains eight integer points. We extend the weight wt(·) : C(λ) → RΩ to a map Ψ λ on the whole space of string parametrisations R l , by using the same definition for real points as for integer points: Ψ λ (u 1 , . . . , u r ) = λ − l ∑ j=1 u j α i j .
Later we shall also consider maps Ψ x with x arbitrary in C; the definition is the same as above, with x = λ. For any dominant weight λ, the map Ψ λ is affine, and (Ψ λ ) |C(λ) = wt. The image of the string polytope P(λ) by Ψ λ is a polytope in RΩ, and one can show that it is the convex hull of the points in W(λ); see for instance [AB04, Definition 1.3]. Moreover, the image by Ψ λ of the Lebesgue measure on P(λ) 1 (u 1 ,u 2 ,...,u l )∈P(λ) du 1 du 2 · · · du l is compactly supported by Conv(W(λ)), and piecewise polynomial (this is a general property of affine images of Lebesgue measures on polytopes); see [BBO09,§5.3]. We can then state a result of asymptotic polynomiality of the Kostka numbers K λ,ω = dim C (V λ (ω)) (instead of the Littlewood-Richarson coefficients): Proposition 7.7. Fix a direction x in the Weyl chamber C ⊂ RΩ, and a continuous bounded function f on RΩ. We assume that x does not belong to the walls of the Weyl chamber. Then, there exists a probability measure m x (y) dy on RΩ that is supported by Conv({w(x) | w ∈ W}), that has a piecewise polynomial density m x , and such that The local degree of y → m x (y) is bounded by l − d = |Φ + | − rank(G), and one has the scaling property m γx (γy) = m x (y) γ d .
The probability measure m x (y) dy is a version of the Duistermaat-Heckman measure, see in particular [BBO09, §5.3].
Proof. Set λ = tx. The left-hand side L( f , x, t) of Equation (15) approximates RΩ f ( ω t ) µ λ (ω), where µ λ is the spectral probability measure of the representation λ, supported on weights and defined by Indeed, the only difference is that we approximated dim C (V λ ) by t |Φ + | ∏ α∈Φ + x | α ρ | α , and this is valid in the limit t → ∞. Now, by the previous discussion, µ λ is the image of the probability measure ν λ = 1 dim C (V λ ) ∑ (n 1 ,...,n l ) integer points in P(λ) δ (n 1 ,...,n l ) on R l by the affine map Ψ λ . Therefore, As t goes to infinity, the discrete measure ν x,t (u) = ν tx (tu) converges in law to the uniform probability mesure υ x on the polytope P(x), which is defined as the set of points of the string cone S (G) which satisfy the inequalities u l ≤ x(α ∨ i l ); u l−1 ≤ (x − u l α i l )(α ∨ i l−1 ); . . . . . . u 1 ≤ (x − u l α i l − · · · − u 2 α i 2 )(α ∨ i 1 ). Therefore, lim t→∞ L( f , x, t) = R l f (Ψ x (u)) υ x ( du). Finally, the image measure m x = (Ψ x ) * (υ x ) is given by a compactly supported piecewise polynomial function, of local degree smaller than l − d; and the obvious identities Ψ γx (γu) = γ Ψ x (u) and γ l υ γx (γu) du = υ x (u) du imply the scaling property. 7.3. From the string polytope to the Littlewood-Richardson coefficients. The theory which enables one to understand the asymptotics of Kostka numbers can be adapted to the same problem with the Littlewood-Richardson coefficients. From the discussion at the end of Section 7.1, and the description of c λ,µ ν as a number of paths in C(π µ ) satisfying certain conditions, one can expect that there is a notion of string polytope of V µ relatively to another dominant weight λ that allows to calculate these coefficients. These relative string polytopes have been constructed by Berenstein and Zelevinsky, see [BZ88,BZ01]. A trail from a weight φ to another weight π of an irreducible representation V µ of g C is a sequence of weights φ = φ 0 , φ 1 , . . . , φ l = π of V µ such that: (1) φ j−1 − φ j = k j α i j for any j ∈ [[1, l]], with the k j 's non-negative integers; (2) there exist in the crystal C(µ) vertices v φ and v π with respective weights φ and π, and a sequence of edges where → k j f α i j stands for k j edges of label f α i j .
In other words, the trails are the images by the weight map of directed paths on the crystal graph. For instance, in the crystal of the adjoint representation of SU(3), there is a trail from ω 1 + ω 2 to ω 1 − 2ω 2 , since one can find the sequence of edges (0, 0, 0) → f α 1 (1, 0, 0) → f α 2 (0, 1, 1) → f α 2 (0, 2, 1) in the crystal graph. We refer to [BZ01, Theorem 2.3] for a proof of the following result, in which we shall consider irreducible representations of the dual Langlands Lie algebra L g C , which is the Lie algebra obtained from g C by exchanging roots and coroots, respectively weights and coweights.
Theorem 7.8 (Berenstein-Zelevinsky). Let S (λ, µ) be the subset of the set of string parametrisations S (µ) that consists of strings (n 1 , n 2 , . . . , n l ) such that, for any i ∈ [[1, d]] and any trail Then, c λ,µ ν is the number of elements with weight ν − λ in S (λ, µ). Therefore, it is the number of integer points in a slice of the Berenstein-Zelevinsky relative string polytope P(λ, µ), which is the intersection of P(µ) with the half-spaces determined by the inequalities above.
of) x, y, z. The scaling property of this map forces then the polynomials to be homogeneous of degree l − d. Finally, the form of the domains of polynomiality comes from the following fact. If one projects by an affine map π a compact polytope P in dimension l to a space of dimension d ≤ l, then the non-empty intersections of images of the faces of the polytope P partition the image π(P) into a finite number of polytopes. On each of these non-empty intersections, the image of the uniform measure on P is polynomial, hence the second part of the proposition.
Proposition 5.7 is the immediate generalisation of Proposition 7.13, and it is proved by applying it recursively and by using the convolution rule for multiple Littlewood-Richardson coefficients, which turns into a convolution rule for the functions q x 1 ,...,x r−1 (z).
Example 7.14. Consider the trivial example where G = SU(2). In this case, a tensor product V kω ⊗ V lω is given by the Clebsch-Gordan rules: The limit when k = tx, l = ty and t → +∞ of this rule is obviously given by the locally constant function q xω,yω (zω) d(zω) = 1 2 1 |x−y|≤z≤x+y dz.
If x = x 1 ω 1 + x 2 ω 2 , then the relative polytope P(x, y) is the subset of P(y) that consists in triplets such that