A comparison of the nonlinear sigma model with general pinning and pinning at one point

We study the nonlinear supersymmetric hyperbolic sigma model introduced by Zirnbauer in 1991. This model can be related to the mixing measure of a vertex- reinforced jump process. We prove that the two-point correlation function has a probabilistic interpretation in terms of connectivity in rooted random spanning forests. Using this interpretation, we dominate the two-point correlation function for general pinning, e.g. for uniform pinning, with the corresponding correlation function with pinning at one point. The result holds for a general finite graph, asymptotically as the strength of the pinning converges to zero. Specializing this to general ladder graphs, we deduce in the same asymptotic regime exponential decay of correlations for general pinning.


Introduction
History of the model. The nonlinear sigma model that we consider here was introduced by Zirnbauer in [Zir91] as a toy model inspired by random matrices in the context of disordered materials. It can be seen as a statistical mechanical model where spins are replaced by vectors with both real and Grassmann components. We associate with each point two real and two Grassmann variables parametrizing a supersymmetric extension of the hyperbolic plane. Therefore the model is often denoted by H 2|2 . In dimension three and higher, a phase transition between a localized (disordered) and an extended (ordered) phase was proved by Disertori, Spencer, and Zirnbauer in [DSZ10] and [DS10].
After integrating out the Grassmann variables in the nonlinear sigma model, the corresponding marginal is a probability measure. It was shown by Sabot and Tarrès in [ST] that this marginal has an interpretation as a mixing measure for a vertex-reinforced jump process and can also be related to linearly edge-reinforced random walk. Exploiting the former relation, the results in [DSZ10] and [DS10] were used by Sabot and Tarrès in [ST] to deduce recurrence for vertex-reinforced jump processes in all dimensions for small initial weights and transience in dimensions d ≥ 3 for large initial weights. For linearly edge-reinforced random walks, Sabot and Tarrès proved recurrence in all dimensions for small initial weights. An alternative proof, without using the connection to H 2|2 , was given by Angel, Crawford, and Kozma in [ACK14]. In dimensions d ≥ 3, Disertori, Sabot, and Tarrès showed in [DST14] transience for linearly edge-reinforced random walks for large initial weights using techniques similar to the one used in [DSZ10]. In [DMR14], we proved recurrence for vertex-reinforced jump processes on general ladder graphs with arbitrary constant initial weights using the connection to H 2|2 just mentioned and a transfer operator method applied to H 2|2 .
Aim of this paper. Without a regularization the model H 2|2 is ill-defined. On a lattice, the most natural choice is to introduce a translationally invariant regularization. This is equivalent to introduce a constant "magnetic field" ε in the corresponding statistical mechanics model. This magnetic field at point j can be interpreted as a "pinning", forcing the spin at point j to remain near a certain value. Then, a constant magnetic field can be seen as uniform pinning. Another possibility is to take an inhomogeneous magnetic field, the easiest choice being pinning at a single point.
In this paper, we consider the model on a finite connected undirected graph G, rather than only lattices. The pinning can also be seen as a modification of the underlying graph as follows. We augment G by an additional vertex ρ in two different ways. In the case of uniform pinning, ρ is connected to every other vertex. In the case of pinning at one point, ρ is only connected to a single vertex in G. When G is a lattice or a ladder graph, the first graph has a nonlocal structure since the graph distance between any two vertices is bounded by 2, whereas the second graph remains local.
In the case of ladder graphs, the local structure for pinning at one point allowed us to prove exponential decay of correlations for arbitrary inverse temperature β; see [DMR14]. However, due to the nonlocal structure of the augmented graph, a similar method is not directly applicable for uniform pinning.
The aim of this paper is to bound the expectation of the Green's function in the case of uniform (or general) pinning with the corresponding Green's function for pinning at one point, asymptotically as ε → 0, for any inverse temperature β > 0. This result holds for general finite graphs. Specializing it down to ladder graphs, it allows us to transfer known bounds for pinning at one point to the case of the Green's function for uniform pinning.
How this paper is organized. In Section 2, the model is formally defined and the results are stated. In Subsection 3.1, we relate the Green's function with a probability concerning certain random spanning trees. Subsection 3.2 contains the proof of the comparison between the different pinnings. The model H 2|2 with pinning at one point has a product structure when passing to gradient variables, that the model H 2|2 with uniform pinning does not exhibit. This product structure is explained in the appendix.

Formal definition
Let G = (V, E) be a finite connected graph with vertex set V and edge set E, consisting of undirected edges i ∼ j. We extend G by adding an extra vertex ρ and edges i ∼ ρ connecting it to every other vertex i. The extended graph is denoted by We attach to every edge i ∼ j of G ρ an edge weight β ij = β ji > 0. In particular, for i ∈ V , the edge i ∼ ρ gets the weight β iρ =: ε i ≥ 0. We assume that ε i > 0 for at least one vertex i ∈ V . To every vertex i ∈ V , we associate two real variables t i and s i , and abbreviate t := (t i ) i∈V and s := (s i ) i∈V . Furthermore, we set t ρ := 0 and s ρ := 0. For i, j ∈ V ρ , we define In the following, we study an equivalent version of the model H 2|2 , where the contribution from Grassmann variables is replaced by a sum over spanning trees T of G ρ . Let T denote the set of spanning trees of G ρ . We identify every tree with its edge set. The spanning trees T of G ρ are in a natural one-to-one correspondence with rooted spanning forests of G as follows. Given T ∈ T , the corresponding spanning forest has the edge set (2.4) Using this notation, we have for t ∈ R V and T ∈ T (2.5) In this representation, H 2|2 is described by the following probability measure on where dt j and ds j , j ∈ V , denote the Lebesgue measure on R and dT is the counting measure on T . As is shown in [DSZ10], using supersymmetry and the equivalent description in terms of Grassmann variables, µ ε is a probability measure.
For convenience, we suppress the dependence of µ ε on G and β in the notation. The symbols t, s, and T are used in two slightly different ways: On the one hand, they denote the canonical projections on R V × R V × T . On the other hand, the same symbols are used as integration (or summation) variables.
Consider the matrix A(t) ∈ R V ×V defined by else. (2.7) Let A ρ (t) denote the analog of A(t) when the underlying graph is G ρ instead of G and let ε be the following diagonal matrix: Deleting the row and the column indexed by ρ from A ρ (t), we get the matrix A ρ (t) ρ c ρ c = A(t) +ε(t). Hence, by the well-known matrix tree theorem (see [Abd04] for a generalized version), Consequently, the (t, s)-marginal of µ ε is precisely the supersymmetric sigma model studied e.g. in [DS10].

Results
The proofs of all results stated here are given in Section 3. Let x, y ∈ V be two different vertices and π i ≥ 0, i ∈ V , be fixed numbers with π x , π y > 0. Given ǫ > 0, we set ε = (ε i ) i∈V = (π i ǫ) i∈V . We are interested in the following Green's function G ε xy := e tx+ty (A(t) +ε(t)) −1 xy . (2.10) It has the following probabilistic interpretation in terms of random weighted spanning trees: Lemma 2.1 There is a version P β,t of the conditional law of µ ε given t that fulfills with β e (t) := β ij e t i +t j for e = (i ∼ j). Writing x T ↔ y if x and y are connected in the spanning tree T through a path which does not use ρ, we have We define O π xy := e tx+ty π x e tx + π y e ty 1 {(x∼ρ)∈T and x T ↔y} . (2.13) As a consequence of Lemma 2.1, one has (2.14) The expression E µ ε [·|t] stands for the conditional expectation given t.
We denote by ε x δ x ∈ R V the vector with coordinate ε x at x and coordinates 0 at all other locations. Our main theorem can now be phrased as follows.
Theorem 2.2 Let π i ≥ 0, i ∈ V , be fixed numbers with π x , π y > 0. We have the following asymptotic comparison between the supersymmetric sigma model with arbitrary pinning ε = (ε i ) i∈V = (π i ǫ) i∈V and with pinning at one point: In particular, the two limits exist.
Remark. It is important that in (2.15) we keep the random variable O π xy with the original π. Indeed, if we replaced O π xy by O πxδx xy we would get by formula (A.7) in the appendix. This bound would give no information on eventual decay in |y − x|.
Ladder graphs. In the special case of quasi-one-dimensional graphs this theorem implies exponential decay of the expectation of G ε xy . More precisely, consider a finite undirected graph G 0 = (V 0 , E 0 ) with vertex set V 0 and edge set E 0 . Let G = (V, E) be the "ladder" built of copies G n = (V n , E n ), n ∈ Z, of G 0 . More precisely, we take the vertex sets V := Z × V 0 and V n := {n} × V 0 , where the copy at level 0 is identified with G 0 . The edge set E consists of "vertical" edges e n := ((n, v) ∼ (n, v ′ )) with n ∈ Z and e = (v ∼ v ′ ) ∈ E 0 and "horizontal" edges v n+1/2 := ((n, v) ∼ (n + 1, v)) with n ∈ Z and v ∈ V 0 . For L, L ∈ N, we set L := (−L, L) and denote by G L the subgraph of G consisting of the vertex set V L := {−L, . . . , L} × V 0 and the edge set E L containing all edges e ∈ E connecting two vertices in V L . We associate with every edge e ∈ E a weight β e > 0. We assume that the family of weights β := (β e ) e∈E is translation invariant in the sense β en = β e 0 and β v n+1/2 = β v 1/2 for all n ∈ Z, e ∈ E 0 , and v ∈ V 0 . For x = (n, v) and y = (m, w) in V , their horizontal distance is defined by |x − y| := |n − m|. Let µ ε L denote the distribution of the supersymmetric sigma model on the graph G L with pinning ε = (ε i ) i∈V L = (ǫπ i ) i∈V L , where π i ≥ 0, i ∈ V L , with at least one π i > 0. Corollary 2.3 There exist constants c 1 , c 2 > 0 depending only on G 0 and β such that for any L, any two different vertices x, y ∈ V L with π x , π y > 0, and c 3 (π) := min{π x , π y } −1 one has For sufficiently small β > 0, the methods from [DS10] can be used to prove a version of (2.17) not only for the limit as ǫ ↓ 0, but also for given ǫ > 0 small enough. In contrast, our result holds for arbitrary β > 0, but only asymptotically for ǫ ↓ 0.
3 Proofs 3.1 Probabilistic interpretation of G ε xy Let G and G ρ be the graphs introduced in Section 2.1. Without loss of generality, we assume V = {1, . . . , n} throughout this subsection. Recall that T denotes the set of spanning trees of G ρ . We will use the definitions (2.3) of the forest F (T ) and (2.4) of the set of roots R(T ). Finally, recall that x T ↔ y iff x and y are connected in F (T ). For any x, y ∈ V , let (3.1) For the proof of Lemma 2.1, we need the following result.
Lemma 3.1 Consider x, y ∈ V and a real symmetric matrix M ∈ R V ×V with M ij = 0 whenever i = j and there is no edge between i and j. Then, the determinant of the minor of M obtained by taking away the column x and the row y is given by Proof. In the case that G is the complete undirected graph with vertex set V , this is the special case of Theorem 1 of Abdesselam's article [Abd04] for a symmetric matrix M when the index sets I and J are replaced by singletons {y}, {x} ⊆ {1, . . . , n}, respectively. Note that the sign ε(F ) appearing in Abdesselam's formula equals 1 in our special case because I = {y} and J = {x} are singletons. Since M ij = 0 whenever i = j and there is no edge between i and j, formula (3.2) remains literally true if we replace the complete graph by the given graph G.
Proof of Lemma 2.1. The tree dependent part of the density of µ ε in (2.6) is given by e∈T β e (t); note that s and T are conditionally independent given t. Consequently, formula (2.11) describes indeed the law of T conditional on t.

Comparing different pinnings
We write the random variable O π xy defined in (2.13) as a sum: The proof of Theorem 2.2 is based on Lemmas 3.2 and 3.4, below, dealing with the first and second summand in (3.9), respectively. Surprisingly, the main mass contributing to the expectation of the first and second summand in (3.9) comes from quite different locations; see also the explanations following formula (3.31), below. On the one hand, values (t, s) with t j ≈ − log ǫ carry most of the mass for the expectation of O π xy 1 {R(T )={x}} . On the other hand, the main contribution to the expectation of O π xy 1 {|R(T )|>1} comes from values (t, s) with t j ≈ + log ǫ. Thus, for small ǫ > 0, in the two expectations the main masses sit at opposite ends. As we shall see, the main contribution comes from the term with precisely one root at x. We examine this contribution first.
The proof will be a consequence of a more general result, where the family ε is replaced by another family ǫa and the random variable O π xy 1 {R(T )={x}} is multiplied by an additional factor χ.
Take a = (a i ) i∈V with a i ≥ 0 for all i and a x > 0 and an additional density function χ : R V → (0, 1]. We will study the expectation where we abbreviate t + log ǫ := (t i + log ǫ) i∈V . The main contribution to the expectation coming from values of the t i 's close to − log ǫ motivates us to shift the t i 's by log ǫ. For this purpose, we introduce new variables: With this definition, s ′ x = 0. Therefore, we will use as new integration variables (t ′ i ) i∈V , s x , and (s ′ i ) i∈V \{x} . For any fixed configuration t ′ , s ′ of these new variables, we consider an auxiliary random variable S ′ on some probability space, taking values s ′ i with probabilities is the normalizing constant. We denote by E a,t ′ ,s ′ and Var a,t ′ ,s ′ the corresponding expectation and variance operators, respectively. In order to have a compact notation, we abbreviate in the following Note that κ depends also on the fixed quantities π, (β ij ) (i∼j)∈E , and on the vertices x, y, although this is not displayed. We remind that s ′ x = 0 by construction.

Lemma 3.3 With all the definitions above, we have
Proof. Note that on the event {R(T ) = {x}}, the root contribution in (2.5) is given by ǫa x e tx . Using (2.6), we get β ij e t i +t j · ǫa x e tx χ(t + log ǫ) e tx+ty π x e tx + π y e ty 1 {R(T )={x} and x T ↔y} . (3.18) Changing variables according to (3.13), we get ( For counting powers of ǫ in the last equality, we have used that F (T ) is a spanning tree of G for |R(T )| = 1 and consequently |F (T )| = |V | − 1. Next we integrate out s x . In terms of the auxiliary random variable S ′ as specified in (3.14), the part of the exponent in (3.19) containing s x can be rewritten as follows: Inserting this into (3.19), we obtain the equality claimed in (3.17).
The ǫ-dependent integrand in (3.17) increases as ǫ ↓ 0. Hence, by the monotone convergence theorem, we get the claimed limit.
To prove the remaining claim (3.11), we compare three special cases of formula (3.17) in Lemma 3.3. Case 1: a = π, χ = 1; i∈V \{x} e − 1 2 ǫπ i e t i ; (3.24) Consequently, the limits in (3.17) for the first two cases coincide, while the limit in the third case yields an upper bound for the other two cases. Hence, Recall that the event {R(T ) = {x}} holds µ ε x δx -almost surely. Consequently, we can drop the indicator function in the last two expectations. Next, we argue that the last limit is finite. Clearly, from (2.13), we have O π xy ≤ min e tx π y , e ty π x . (3.26) By formula (A.7) in the appendix, we conclude (3.27) Since the upper bound is independent of ǫ, we have the same bound for the limit: lim ǫ↓0 E µ εxδx O π xy ≤ min{π −1 x , π −1 y }. The next lemma deals with the lower order corrections coming from forests with at least two roots. Proof. Let S be a fixed spanning tree of G. We drop the interaction terms β ij (B ij −1) ≥ 0 along the edges (i ∼ j) ∈ S ∪ {x ∼ ρ}. This yields ε i e t i · e tx+ty π x e tx + π y e ty 1 {(x∼ρ)∈T and x T (3.29) In the following, we first change variables to s x and gradient variables s ij := s i − s j , (i ∼ j) ∈ S, along the spanning tree S, where the edges in S are oriented in a fixed, but arbitrary way. Since S is a spanning tree, this is a well defined coordinate change. Then we integrate the new variables out.
r.h.s. in (3.29) = (3.30) Next, we set for i ∈ V . In particular, τ x = 0; thus, we use t ′ x and τ i , i ∈ V \ {x}, as new integration variables. Note that this substitution is different from the one in the proof of Lemma 3.2. Heuristically speaking, the reason is that in the case of |R(T )| > 1 most of the mass of the t x -integral is located near t x ≈ + log ǫ, while in the case of one root R(T ) = {x} the mass is essentially located near t x ≈ − log ǫ. To do the power counting for ǫ and e t ′ x in the following calculation, we use |F (T )| + |R(T )| = |V ρ | − 1 = |V | = |S| + 1. (3.32) We obtain Note that in this integral, the integrand decays superexponentially for t ′ x near −∞ and exponentially for t ′ x near +∞. Thus, we get π i e τ i · 1 π y =:ǫe ǫπx c 5 (π, β, G). (3.35) Note that c 5 (π, β, G) < ∞ because the product over e −β ij [cosh(τ i −τ j )−1] decays superexponentially fast (recall that τ x = 0). Summarizing, we get The main theorem 2.2 is now proved by a combination of Lemmas 3.2 and 3.4: Proof of Theorem 2.2. From (3.9), we get (3.37) Combining this with Lemma 3.2 and Lemma 3.4 yields (3.38) Using (2.14), we obtain (3.39) Applying (3.38) twice, as it is and with x and y interchanged, the claim follows. Finally, specializing the theorem down to ladder graphs, we transfer our results from [DMR14] concerning exponential decay of weights in the case of pinning at one point to the case of uniform pinning (or more general pinning): Proof of Corollary 2.3. Recall that c 3 = min{π x , π y } −1 . We estimate O π xy ≤ e tx+ty π x e tx + π y e ty ≤ c 3 min{e tx , e ty } ≤ c 3 e tx e 1 4 (ty−tx) . (3.42) By Theorem 2.1 in [DMR14], there exist constants c 6 , c 7 > 0 depending only on G 0 and β such that Interchanging the roles of x and y, we get the same upper bound for lim ǫ↓0 E µ εy δy ]. An application of Theorem 2.2 yields the claim.
A Appendix: Product structure of the model with single pinning When transforming the model H 2|2 with pinning at one point to gradient variables, it exhibits a certain product structure coming from scaling symmetry. This is made precise in the following lemma.
Lemma A.1 With respect to µ εxδx , the random pair (t x , s x ) is independent of the random vector consisting of the (rescaled) gradient variables The joint distribution of (t x , s x ) with respect to µ εxδx has the density ε x 2π exp −ε x cosh t x − 1 + 1 2 s 2 x e tx , (A.2) independently of the graph G. In particular, The joint distribution of (t ′ i , s ′ i ) i∈V \{x} does not dependent on ε x .
Proof. Recall the definition of µ ε given in (2.6). In the special case ε = ε x δ x , the random tree T contains µ εxδx -almost surely the edge x ∼ ρ, but no other edge of the type i ∼ ρ, i = x. Hence, we get µ εxδx (dt ds dT ) = j∈V dt j ds j e −t j 2π dT e −εx(Bxρ(t,s)−1) ε x e tx Let ν εxδx denote the joint distribution of (t x , s x , (t ′ i , s ′ i ) i =x ). We set t ′ x := 0 and s ′ x := 0. Note that (s i − s j ) 2 e t i +t j = (s ′ i − s ′ j ) 2 e t ′ i +t ′ j . Changing variables accordingly and denoting the set of spanning trees of the graph G by T G , we obtain In the special case of the graph G consisting of only one point x, i.e. V = {x} and E = ∅, the measure ν εxδx has the density given in (A.2). Since ν εxδx is a probability measure, the density in (A.2) is normalized to have total mass one. Consequently, given the product structure in (A.5), for a general graph G, the random vectors (t x , s x ) and (t ′ , s ′ ) are independent with the claimed first marginal and the second marginal not depending on ε x . Finally, we calculate