Characteristic polynomials of real symmetric random matrices

TL;DR

This paper demonstrates the universal local statistical behavior of characteristic polynomials of real symmetric random matrices in the large size limit, using dual representations and integral formulas, with implications for external source models.

Contribution

It introduces a dual integral representation for correlation functions of real symmetric matrices and extends the Itzykson-Zuber method to this setting, including finite corrections.

Findings

Correlation functions exhibit universal local statistics at large N.

A dual representation involves integrals over quaternionic matrices.

Explicit formulas for matrices with external sources are derived.

Abstract

It is shown that the correlation functions of the random variables $\det(\lambda - X)$, in which $X$ is a real symmetric $ N\times N$ random matrix, exhibit universal local statistics in the large $N$ limit. The derivation relies on an exact dual representation of the problem: the $k$-point functions are expressed in terms of finite integrals over (quaternionic) $k\times k$ matrices. However the control of the Dyson limit, in which the distance of the various parameters $\la$'s is of the order of the mean spacing, requires an integration over the symplectic group. It is shown that a generalization of the Itzykson-Zuber method holds for this problem, but contrary to the unitary case, the semi-classical result requires a {\it finite} number of corrections to be exact. We have also considered the problem of an external matrix source coupled to the random matrix, and obtain explicit integral formulae, which are useful for the analysis of the large $N$ limit.