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

Y V Fyodorov,A Nock

doi:10.1007/s10955-015-1209-x

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.

Highlights

The heuristic procedure is widely employed in Theoretical Physics for random matrix theory (RMT) applications to Quantum Chaos using the properties of the standard Gaussian integrals over complex or real variables
In this paper we have started the program of systematic evaluation of correlation functions (1) involving half-integer powers of the characteristic polynomials of N × N Gaussian orthogonal ensemble (GOE) matrices
The method in a nutshell amounts to replacing the initial average involving the product of K characteristic polynomials divided by L square roots of characteristic polynomials of N × N GOE matrices H with an average over the sets of K × K matrices Q F and L × L matrices Q B > 0 with Gaussian weights augmented essentially with the factors det Q B and det Q F raised to powers of order N, see e.g. (35)

Summary

Introduction

Det(μF1 − H ) . . . det(μF K − H ) det1/2(μB1 − H ) . . . det1/2(μBL − H ) G O E (1). We start with considering correlation functions with two square roots in the denominator, and with one or two characteristic polynomials in the numerator, that is C1,2(μF1; μB1, μB2) and C2,2(μF1, μF2; μB1, μB2), and treat a special case of the correlation function involving four square roots in the denominator, and two determinants in the numerator, that is C2,4 in our notation As it should be clear from the examples given below the most physically interesting (bulk) scaling regime in the large-N limit arises when all spectral parameters are close to some value E ∈ (−2J, 2J ) by a distance of the order of the mean spacing between neighbouring eigenvalues in the bulk, i.e. O(J/N ). Our methods are tailored for dealing with the GOE we expect our results in the bulk scaling limit to be universal and shared by a broad class of invariant measures on real symmetric matrices H [8] and by so-called Wigner ensembles of random real symmetric matrices with independent, identically distributed entries satisfying relevant moments conditions [9,10]

Motivations and Background

The Results

Derivation of the Main Results

Conclusions and Open Problems