SUSY transfer matrix approach for the real symmetric 1d random band matrices

Tatyana Shcherbina

doi:10.1214/22-ejp747

Abstract

This paper adapts the recently developed rigorous application of the supersymmetric transfer matrix approach for the Hermitian 1d band matrices to the case of the orthogonal symmetry. We consider 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 Jjk=W−1(δj,k+βΔj,k) in each block. Considering the limit W,n→∞, we prove that the behaviour of the second correlation function of characteristic polynomials of such matrices in the bulk of the spectrum exhibit a crossover near the threshold W∼ N.

Highlights

Starting from the works of Erdos, Yau, Schlein with co-authors and Tao and Vu, significant progress in understanding of universal behaviour of local eigenvalues statistics of many random graph and random matrix models were achieved
The conjecture states that the eigenvectors of N × N random band matrices (RBM) are completely delocalized and the local spectral statistics governed by the Wigner-Dyson statistics for large bandwidth W, and by Poisson statistics for a small W
In this paper we want to perform the complete study of characteristic polynomials for real symmetric Gaussian 1d RBM adapting the supersymmetry techniques (SUSY) transfer matrix techniques of [30], [32] to the case of orthogonal symmetry

Summary

Introduction

Starting from the works of Erdos, Yau, Schlein with co-authors (see [16] and reference therein) and Tao and Vu (see, e.g., [35]), significant progress in understanding of universal behaviour of local eigenvalues statistics of many random graph and random matrix models were achieved. N. In this paper we want to perform the complete study of characteristic polynomials for real symmetric Gaussian 1d RBM adapting the SUSY transfer matrix techniques of [30], [32] to the case of orthogonal symmetry. In this paper we want to perform the complete study of characteristic polynomials for real symmetric Gaussian 1d RBM adapting the SUSY transfer matrix techniques of [30], [32] to the case of orthogonal symmetry This is an important step towards the proof of the universality of the usual correlation functions for the case of real symmetric 1d RBM, as well as for the general development of rigorous application of SUSY approach for the real symmetric case.

Notation

Integral representation

Representation in the operator form

Saddle-point analysis

Findings

A SUSY techniques