On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

TL;DR

This paper evaluates large-N limits of correlation functions involving half-integer powers of GOE characteristic polynomials using supersymmetry, with applications to quantum chaos and wave scattering.

Contribution

It provides explicit formulas for these averages and derives the distribution of off-diagonal resolvent entries in GOE matrices, advancing analytical tools in RMT.

Findings

Explicit large-N limit formulas for correlation functions

Distribution of off-diagonal resolvent entries derived

Applications to chaotic wave scattering experiments

Abstract

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian Orthogonal Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large-$N$ limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.