MODERATE DEVIATIONS FOR TRACES OF WORDS IN A MULTI-MATRIX MODEL

We prove a moderate deviation principle for traces of words of weakly interacting random matrices deﬁned by a multi-matrix model with a potential being a small perturbation of the GUE. The remarkable strength of high order expansions of the matrix model recently found by Guionnet and Maurel-Segala is the key fact that allows us to develop our result and provides also an alternative proof for a special case of the central limit theorem for traces of words, studied in [ 11 ] .


The multi-matrix model
In this note we study Hermitian matrices whose distribution is given by a small convex perturbation of the Gaussian Unitary Ensemble, denoted by GUE. We fix m ∈ the number of random matrices we shall consider. Let N ( ) m be the set of m-tuples A = (A 1 , . . . , A m ) of N × N hermitian matrices, such that eA i (kl), k < l, mA i (kl), k < l, 2 − 1 2 A i (kk) is a family of independent real Gaussian variables of variance (2N ) −1 . We will consider the following perturbation of the GUE: where Z N V t is the normalizing constant, V t (A 1 , . . . , A m ) := n i=1 t i q i (A 1 , . . . , A m ) a polynomial potential with n ∈ , t = (t 1 , . . . , t n ) ∈ n and monomials (q j ) 1≤ j≤n fixed, and the law of the m-dimensional GUE with d N A the Lebesgue measure on N ( ) m . In this note we will prove a moderate deviation principle (MDP) and a central limit theorem (CLT) for normalized traces of a non-commutative monomial, tr(q l (A 1 , . . . , A m )), under µ N V t . For m = 1 and V t (A) = 0 we recover the classical GUE. In this case, a MDP for the difference of the cumulative distribution function (cdf) of the semicircle distribution and the cdf of the eigenvalues was proved in [4]. Matrix models, in which general polynomials V t (A) were allowed in (1) while still keeping m = 1, have been studied intensively in physics. The choice V t (A) = t A 4 , t ∈ , was a commonly studied object, see [1,2]. What was really striking, is the connection their analysis established between matrix integrals like Z N V t and map enumeration. A genus expansion of Z N V t Z N and log Z N V t Z N respectively in powers of N was given for V t (x) = 1 2 x 2 + ν k=1 t k x k with suitable chosen parameters (t 1 , . . . , t ν ) and it was placed on solid mathematical grounds by Ercolani and McLaughlin [6], who proved their results via a Riemann-Hilbert approach. It turned out, that the coefficients of the powers of N in the expansion [6, Theorem 1.1] are closely related to map enumeration. Moreover, for m = 1 and for a polynomial potential V t of even degree and with a positive leading coefficient, Johansson [13] proved a CLT for N i=1 f (λ i ), where λ i are the eigenvalues of the hermitian matrix, for functions f fulfilling some kind of regularity condition, being pointwise bounded by C(V t + 1) for a constant C > 0 and its derivative being pointwise bounded by a polynomial. Since many interesting models like the Ising model on random graphs, the q-Potts model on random graphs, the Chain model and the so-called induced QCD model (quantum chromodynamics, see [15]) are of form (1) and have m > 1, the question arose, whether a genus expansion for this multi-matrix model with general polynomial potentials V t could be obtained. A first order expansion was obtained by Guionnet [7], a paper that relied on [12]. Guionnet and Maurel-Segala refined the expansion up to second order [11] until Maurel-Segala [16] gave the full genus expansion for the multi-matrix model of form (1). In [11], the authors also derived a CLT for N times the difference of the empirical measure and its limit, the solution of a Schwinger-Dyson equation. It was also shown that the CLT still holds, when the limit of the empirical measure is replaced by its expectation. For a special case of that CLT, we will give an alternative proof. Finally, notice that Cabanal-Duvillard [3] introduced a stochastic calculus approach and proved a CLT for traces of non-commutative polynomials of Gaussian Wigner and Wishart matrices, as well as for traces of non-commutative polynomials of pairs (m = 2) of independent Gaussian Wigner matrices. After introducing a technical assumption for the polynomial potential V t in (1), we will state our main result in the next subsection. Since the results of the paper [16] are the crucial foundation of this note, we will give the two main theorems of [16] in section 2, in which we finally prove our main result. In the last section, we compare the two different admissible regions of parameters of the polynomial V t as they appear in [6] and [16] and finally, we will apply our result to the Ising model and the q-Potts model.

Main results
Before stating our main result, we introduce the notion of c-convexity for the potential V t .
i is real and convex as a function of the entries of the matrices, we say that V t is c-convex.
If V t is c-convex, the Hessian of the trace of V t (A 1 , . . . , A m ) + 1 2 m i A 2 i is symmetric and positive definite with eigenvalues bigger than c for any N ∈ . Remark, that the condition implies that Z N V t is automatically finite. An example of a c-convex V t is with convex real polynomials P i in one unknown and for all l, k |β k,l | ≤ (1 − c). This is due to Klein's Lemma [9, Lemma 6.2] which states that the trace of a real convex function of a selfadjoint matrix is a convex function as function of the entries of the matrix. Hence, a special class of examples are V t of the form with non-negative t i 's, integers p i 's and real α's (see the potentials V t considered in [6] in the one matrix case m = 1). Taking the potential V t : . . , A m ), n ∈ , with complex t = (t 1 , . . . , t n ) ∈ n and non-commutative monomials (q j ) 1≤ j≤n , we next define for any η > 0 and c > 0 Let denote the expectation with respect to the probability measure µ N V t , defined in (1). Besides being c-convex, we require the potential function to also be self-adjoint in terms of the following definition. Definition 1.2. We say that the potential V t is self-adjoint, if V t = V † t holds with respect to the involution † that is given for all z ∈ and all monomials q l (H 1 , . . . , H m ) = H l 1 . . . H l p by Thus, if the potential V t is self-adjoint, all the appearing monomials q l are also self-adjoint. Note that a self-adjoint potential V t or monomial q l always has a real trace. We say that a sequence of probability measures (µ n ) n∈ , on some topological space X obeys a large deviation principle with speed a n and good rate function I(·) : X → + 0 if • I is lower semi-continuous and has compact level sets N L := {x ∈ X : I(x) ≤ L}, for every L ∈ [0, ∞). Similarly, we will say that a sequence of random variables (Y n ) n∈ with topological state space X obeys a large deviation principle with speed a n and good rate function I(·) : X → + 0 if the sequence of their distributions does. Formally a moderate deviation principle is nothing but an LDP. However, we will speak about a moderate deviation principle (MDP) for a sequence of random variables, whenever the scaling of the corresponding random variables is between that of an ordinary Law of Large Numbers and that of a Central Limit Theorem. Now our result is as follows. For a non-commutative monomial q l , we define where the expectation is with respect to µ N V t .
N obeys a MDP with speed N 2γ and rate function for any 0 < γ < 1. The function F 0 (·) is given by (3).
Since F 0 (t) is the connection between matrix models and map enumeration, we will give some more details (for a detailed explanation of that connection see [19]). A map is a connected graph drawn on a compact oriented connected surface, such that the edges are not intersecting (do not cross each other), the number of holes in the surface in order to avoid intersections is minimal, which is equivalent to obtaining a disjoint union of sets, where each set (or face) is homeomorphic to open disks after cutting the surfaces along the edges. That minimal number of holes is called the genus g of the surface. Thus a planar graph can be drawn on a surface of genus 0. We consider maps that are colored in m colors, i.e. each edge has a color c ∈ {1, . . . , m}.
For a non-commutative monomial in m indeterminants, we define vertices, that are of type q l as follows: We say that a vertex is of type q l , if it has k l colored half-edges, one marked half edge and an orientation, such that the marked half-edge is of color l 1 , the next one with respect to the orientation is of color l 2 and so forth, until the last half-edge is colored with l k l . Thus, we obtain a bijection between monomials and stars. Moreover, the graphical interpretation of the involution † as defined in (2) is quite simple. Just shift the marker of the first half-edge towards the next neighboring half-edge against the orientation of the vertex and afterwards reverse the orientation of the vertex. The function F 0 (t), which appears in the rate function of the MDP, is a generating function for maps of genus 0 associated with V t , being the number of maps on a surface of genus 0 with k i vertices of type q i . It is not quite obvious, how the second derivative of F 0 (t) with respect to t l looks like and we will see below (in Theorem 2.2) that it can be regarded as a generating function for maps of genus 0, which have two fixed stars q l : where j = (0, . . . , 0, j l = 2, 0, . . . , 0)) and everything else as above. Now, we also state a special case of the CLT [11, Theorem 4.7] of which we will give an alternative proof.

Proof of main results
Before we give the proofs, we state the two main theorems of [16], on which the proofs build upon. In physics it is known that the perturbed GUE is related to the enumeration of graphs on surfaces. The main result in [16] is, that for small convex perturbations, the moments of the empirical measure can be developed into a series whose g-th term is a generating function of graphs on a surface of genus g: where F g is the generating function for maps of genus g associated with V t , and k g is the number of maps on a surface of genus g with k i vertices of type q i . Now it is obvious, why such an expansion is called genus expansion and why the leading term F 0 (t) was called the planar approximation. For our purposes, we will only apply that theorem for g = 1.
That the expansion in (5) can be derived term by term is the content of the next theorem. In fact, this means that the asymptotics of much more observables are available. For j=( j 1 , . . . , j n ) ∈ n , we introduce the operator of derivation Theorem 2.2 (Theorem 1.3 in [16]). Let V t as above, c > 0. For all j = ( j 1 , . . . , j n ) ∈ n , for all g ∈ , there exists η g > 0, such that for all t ∈ B η g ,c , where j F g (t) is the generating function for rooted maps of genus g associated with V t with some fixed vertices: where k g is again the number of maps on a surface of genus g with k i vertices of type q i . For details see [16,Theorem 7.4].
Thus, we find for example for m = 2, V t = t 1 Theorems 2.1 and 2.1 yield asymptotic information concerning the statistics of words of the multimatrix models. For example, by differentiating log In conjunction with (6), one learns the following: Out of these two theorems, we deduce an asymptotic expansion for the moment generating function of φ l := φ l (A) := tr(q l (A 1 , . . . , A m )) and φ l : Remember, that denotes the expectation with respect to the probability measure µ N V t . Lemma 2.3. Let t ∈ n and c > 0. Then there exists η > 0, such that for self-adjoint V t = n i=1 t i q i (A 1 , . . . , A m ), n ∈ , with t in B η,c ∩ n , and any fixed l ∈ {1, . . . , n} we have for any s ∈ that In the non-centered case, we find Proof. Consider for any s ∈ and l ∈ {1, . . . , n} the potential V t− s N e l . We easily see that We abbreviate V := V (t, s, l) := V t− s N e l and obtain We want to apply Theorem 2.1 for both terms on the right hand side of the last equality. Fix c > 0. Then by Theorem 2.1 there exists a η := η(c) > 0 such that the expansion (5) holds true for Z N V t for all t ∈ B η,c . Now for fixed s ∈ we choose N sufficiently large such that |s/N | < for a > 0. We abbreviate t := t − s N e l . For any t ∈ B η,c the polynomial V t is c-convex, thus for N sufficiently large we can find a c > 0 such that V t is c -convex. For this c > 0 we can find a η := η (c ) > 0 such that (5) holds true for any t ∈ B η ,c , where V t is c -convex. Now we choose < η and η := min(η − , η). Since η ≤ η, the condition |t| ≤ η induces |t| ≤ η and therefore, the relevant set to consider is B η ,c . Summarizing, for fixed s ∈ and N sufficiently large, for any t ∈ B η ,c we can apply Theorem 2.1 for both terms in (8). Remark that we apply Theorem 2.1 taking g = 1. Now we obtain the following asymptotic expansion in the non-centered case: In the centered case, we also have to consider the contribution of exp(−s [φ l (H)]) to the asymptotic expansion. As it is well known in statistical mechanics and as we have seen in (7), differentiating the free energy yields the expectation of the observables and hence we find Finally, (8), (9) and Theorem 2.2 yield and therefore we obtain This lemma is now the crucial ingredient for the proofs of our main results.
Proof. (Proof of the central limit theorem, Theorem 1.4) By Taylors theorem, we have where Combining this with Lemma 2.3, we find the limit of the moment generating function of φ l to be that of a centered random variable having a normal distribution with variance ∂ 2 since ξ 0 N → t, for N → ∞, and F 0 , F 1 being differentiable functions of any order around the origin by Theorem 2.2 (they are even analytic functions, see [16,Lemma 4.4]). By Levy's continuity theorem the CLT is established.
Proof. (Proof of the moderate deviation principle, Theorem 1.3) In order to apply the Gärtner-Ellis theorem (cf. Theorem 2.3.6 in [5]), we calculate the limit of the properly scaled cumulant generating function of the random variable 1 N γ φ l , where 0 < γ < 1: let s ∈ , we obtain The second equality is due to Lemma 2.3. Although we cannot simply replace s by sN γ and assume the lemma to hold, we can carefully go through the proof of Lemma 2.3 when the new scaling is applied. It turns out, that the lemma still holds, which basically relies on N γ /N going to 0. The third equality is due to Taylor's theorem (10) and the last one follows by F 0 being differentiable of any order around the origin and the fact that ξ 0 N ∈ t ∧ (t − sN γ N e l ), t ∨ (t − sN γ N e l ) , which yields ξ 0 N → t, for N → ∞, since γ ∈ (0, 1). In particular the right hand side of (11) is finite for all s ∈ , is everywhere differentiable in s and steep, since Thus, we can apply the Gärtner-Ellis theorem, and finish the proof by calculating the Legendre-Fenchel transform of s 2 ∂ 2 ∂ t 2 l F 0 (t), which is

Remarks:
(1) For all the calculations carried out above, the second order expansion (g=1) in [11] could have been sufficient, while we needed Theorem 2.2 to establish our Lemma 2.3, which was crucial for proving our result.
(2) Instead of using the asymptotic expansion of log Z N V t up to order g = 1 in Lemma 2.3 and applying Taylor's theorem, we could have only used that expansion up to order g = 2 for proving the MDP and the CLT.
(3) In case of m = 1, we would find e.g. the limiting variance of φ l to be ∂ 2 where κ k+2e l 0 is the number of maps on a surface of genus 0 with k i vertices of valence i, i = {1, . . . , n}, and two vertices of valence l. That corresponds to a non-colored graph, respectively a graph in which all edges are colored with one color. Notice moreover that F 0 (t) corresponds to e 0 (t 1 , . . . , t n ) in the notation of [6].
? Along the proof of (11) we obtain that for s ∈ , such that t − se l ∈ B η,c . Hence we are only able to apply the Gärtner-Ellis theorem locally, for sufficient small s, obtaining a LDP only locally for sufficiently small intervals (−ε, ε) with a implicit rate function In other words, the expansions in [16] are not strong enough to obtain a full LDP. For a similar discussion in case of m = 1, see [17]. Generalization: All the results above can be generalized from monomials to polynomials, n i=1 α i tr(q i ), provided that each q i has a non-vanishing coefficient t i in the potential V t . We will briefly state the crucial steps of the calculations and only look at ψ = α 1 tr(q l 1 )+α 2 tr(q l 2 ), where l 1 , l 2 ∈ {1, . . . , n}, l 1 = l 2 , α 1 , α 2 ∈ fix, and ψ = α 1 tr(q l 1 ) + α 2 tr(q l 2 ) − [α 1 tr(q l 1 ) + α 2 tr(q l 2 )] respectively. In order to be able to apply Theorem 2.1 and a Taylor expansion in two variables, we argue as in the proof of Lemma 2.3 that for c > 0 we can find a η = η(c) > 0 such that t −κ s N (α 1 e l 1 +α 2 e l 2 ) ∈ B η,c for every κ ∈ [0, 1] and N sufficiently large. With the new potential V t− s N (α 1 e l 1 +α 2 e l 2 ) = V t − s N (α 1 q l 1 + α 2 q l 2 ), we find that e s(α 1 trq l 1 +α 2 trq l 2 ) = Z N V t− s N (α 1 e l 1 +α 2 e l 2 ) Centering ψ, we need to calculate [α 1 trq l 1 + α 2 trq l 2 ]. As above, the following holds, The moment generating function now turns out to be (via Theorem 2.1) e sψ Hence we obtain and therefore Here, we used the Taylor expansion in two variables and ξ N = t − κ s N (α 1 e l 1 + α 2 e l 2 ), for a κ ∈ (0, 1). Having thus obtained the CLT for ψ under µ N V t , we can use the expansion (14) to obtain the MDP for 1 where γ ∈ (0, 1), with speed N 2γ and rate function via the Gärtner-Ellis approach: , with κ ∈ (0, 1) and F 0 being differentiable of any order for t sufficiently small. Since s 2 is finite for s ∈ , everywhere differentiable in s and steep, we have established the MDP.

Discussion and Applications
Let us consider the one-matrix model m = 1 first. In this case, the results of [16] are comparable to the expansions given in [6, Theorem 1.1]. The former results can be applied to V t being a c-convex polynomial with t ∈ B η,c ∩ n . The latter results hold true for polynomials V t such that t = (t 1 , . . . , t n ) lies in the region To be more specific, there is a T > 0 and a γ > 0 such that for t ∈ (T, γ) the asymptotic expansion of log Z N V t given in [6, Theorem 1.1] is available. An analytic comparison between these two admissible regions of parameters prompts us to claim the following : Claim 1: If n is even and T > 0 and γ > 1 are such that T < 1 n(n−1) 2 (1+γ) , it holds that (T, γ) ⊂ B T,c(T,γ) for some c(T, γ) > 0.
In the situation of claim 1, the expansion of log Z N V t does not hold a priori as we did not choose T and γ with respect to that condition but rather arbitrary. But we can deduce that for our choice of T and γ as above, the expansion of log Z N V t holds for any t ∈ (T, γ), because the potential can be shown to be c-convex. Because of this observation, it is sometimes mentioned that the c-convexity encompasses the condition of Ercolani and McLaughlin for T sufficiently small and γ sufficiently large, see e.g. [9]. While the first claim only holds for even n, the second claim states that for uneven n, only the expansion of Ercolani and McLaughlin is applicable, since V t cannot be c-convex in this situation.
Proof. (Proof of Claim 1) Let T > 0 and γ > 1 be as in claim 1. As m = 1, the potential V t is of is always real. We will deduce the existence of a c(T, γ) such that V t is c-convex by an application of Klein's lemma and therefore we show that f (x) := n i=1 t i x i + 1−c 2 x 2 is a convex function. Thus, we need to establish the positivity of the second derivative, g(x) := f (x) = n i=2 t i (i − 1)i x i−2 + 1 − c ≥ 0 for any x ∈ . Observe, that we can find a c 1 (T, γ) such that g(0) = 2t 2 + 1 − c > 0, because of γ > 1 and T < 1 2 for n ≥ 2. The next case to consider is that of |x| ≥ 1. As t ∈ (T, γ), it is obvious from (16) that Because n is even and γ > 1, it holds for every i ≤ n that γ|t i |(n − 1)n x n−2 = γ|t i |(n − 1)n x n−2 > t i (i − 1)i x i−2 , which gives g(x) > 0 for x with |x| ≥ 1. When |x| < 1, we start with the observation that 0 ≤ n−1 i=1 t i (i − 1)i x i−2 + γ|t i |(n − 1)n x n−2 < T (n − 1) 2 n + γT n(n − 1) 2 < 1.
Proof. (Proof of Claim 2) In case of uneven n, take e.g. the potential V t = 2t 2 H 2 +t 3 H 3 . Obviously, this potential is not c-convex (take e.g. the case N = 1, in which the matrix H reduces to one realvalued unknown h and we immediately see that tr V t (h) + 1−c 2 h 2 = 4t 2 +1−c 2 h 2 + t 3 h 3 is not a convex function in h for any c > 0, regardless of our choice of η > 0). Along the same lines, we see that any potential V t with t n = 0 for n uneven cannot be c-convex.

Remark:
When using the expansion (5), a CLT and a MDP for the centered and properly scaled random variable tr(q l ) = tr(X l ) could only be obtained if t l = 0. Considering the case of V t = t n H n , n ≥ 2, either even or uneven, we see that the expansion can be applied to the numerator and denominator in (8) even for t l with l < n: Provided that t ∈ o (T, γ), where o (T, γ) denotes the interior of the set (T, γ), it still holds for large N that max{t n , | s N |} < T and t n > |γ s N |, for s ∈ . And once this expansion was established, the proofs can be left unchanged to yield the CLT and MDP for the distribution of the centered and properly scaled random variable tr(X l ) =