Universality for 1d Random Band Matrices: Sigma-Model Approximation

Abstract

We consider 1d random Hermitian N × N block band matrices consisting of W × W random Gaussian blocks (parametrized by j, k ∈ Λ = [1, n] ∩ Z, N = nW ) with a fixed entry's variance J jk = W −1 (δ j,k + β∆ j,k ) in each block. Considering the limit W, n → ∞, we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as W ≫ √ N , is determined by the Wigner -Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in [20].

Delocalization Transition for Critical Erdős–Rényi Graphs

“…They are conjectured [40] to have a similar phase diagram as the Anderson model in dimensions n 3. As for the Anderson model, dimensions n > 1 have so far seen little progress, but for n = 1 much has been understood both in the localized [48,50] and the delocalized [20][21][22][28][29][30][31][32][33]43,51,52,59] phases. A simplification of band matrices is the ultrametric ensemble [41], where the Euclidean metric of Z n is replaced with an ultrametric arising from a tree structure.…”

Transfer Matrix Approach to 1d Random Band Matrices: Density of States

“…However for the Hermitian RBM of a certain type it was successfuly done both for correlation functions of characteristic polynomials and for usual correlation functions. More precisely, combining SUSY with a delicate steepest descent method and transfer matrix techniques, we were able to perform a complete study of the local regime of characteristic polynomials for Hermitian Gaussian 1d RBM (see [27] for the regime W ≫ √ N , [28] for the regime W ≪ √ N , and [30] for the regime W ∼ √ N ), and also to obtain the first rigorous universality result for the second order correlation function for the whole delocalized region W ≫ √ N (see [29]). There are much less rigorous application of SUSY techniques for the case of real symmetric matrices, since the SUSY integral representations are more complicated for the case of orthogonal symmetry.…”

“…The model we are going to consider is different from the model of 1d RBM considered in [27] - [28], [30] and in [31], but coincides with the model considered in [29]. Namely, we consider real symmetric block band matrices, i.e.…”

Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

“…This produces two terms; by the rapid decay of g there are no boundary terms. For the first term produced by the integration by parts, using (26)…”