Wigner-Dyson Statistics from the Replica Method

TL;DR

This paper uses the fermionic replica method to compute eigenvalue correlations in the Gaussian unitary ensemble, emphasizing the importance of including replica symmetry-breaking saddle points for accurate results.

Contribution

It demonstrates that incorporating replica symmetry-breaking saddle points is essential for correct eigenvalue correlation calculations using the replica method.

Findings

Replica symmetry-breaking saddle points are necessary for accurate correlation functions.

The fermionic replica method reproduces exact large-distance eigenvalue correlations.

Non-trivial saddle points impact the replica calculation's accuracy.

Abstract

We compute the correlation functions of the eigenvalues in the Gaussian unitary ensemble using the fermionic replica method. We show that non--trivial saddle points, which break replica symmetry, must be included in the calculation in order to reproduce correctly the exact results for the correlation functions at large distance.