Random matrices and the replica method

TL;DR

This paper demonstrates that both bosonic and fermionic replica methods can accurately reproduce nonperturbative spectral fluctuation formulas in invariant non-Gaussian random matrix ensembles across the entire energy spectrum without relying on sigma-model mappings.

Contribution

It shows the applicability of replica methods to non-Gaussian ensembles and spectral fluctuations without sigma-model mapping, extending previous theoretical approaches.

Findings

Replica methods reproduce spectral correlation functions nonperturbatively.

Both bosonic and fermionic replicas are effective.

Results apply to invariant non-Gaussian ensembles with various symmetries.

Abstract

Recent developments [Kamenev and Mezard, cond-mat/9901110, cond-mat/9903001; Yurkevich and Lerner, cond-mat/9903025; Zirnbauer, cond-mat/9903338] have revived a discussion about applicability of the replica approach to description of spectral fluctuations in the context of random matrix theory and beyond. The present paper, concentrating on invariant non-Gaussian random matrix ensembles with orthogonal, unitary and symplectic symmetries, aims to demonstrate that both the bosonic and the fermionic replicas are capable of reproducing nonperturbative fluctuation formulas for spectral correlation functions in entire energy scale, including the self-correlation of energy levels, provided no sigma-model mapping is used.