Ergodicity of Random-Matrix Theories: The Unitary Case

TL;DR

This paper proves the ergodicity of unitary random-matrix theories by demonstrating the decay of autocorrelation functions using supersymmetry methods and saddle-point analysis.

Contribution

It introduces a rigorous proof of ergodicity for unitary random-matrix theories employing Efetov's supersymmetry approach and boundary term analysis.

Findings

Autocorrelation functions vanish asymptotically.

Ergodicity holds for unitary random-matrix ensembles.

Methodology involves supersymmetry and saddle-point techniques.

Abstract

We prove ergodicity of unitary random-matrix theories by showing that the autocorrelation function with respect to energy or magnetic field strength of any observable vanishes asymptotically. We do so using Efetov's supersymmetry method, a polar decomposition of the saddle-point manifold, and an asymptotic evaluation of the boundary terms generated in this fashion.