Random Matrix Theory by the Supersymmetry Method Beyond the Sigma-Model

TL;DR

This paper extends the supersymmetry method in Gaussian Random Matrix Theory beyond the sigma-model to accurately compute leading corrections to correlation functions at large and small energy differences.

Contribution

It introduces an extension of the supersymmetry method that includes quadratic fluctuations, improving the precision of correlation function calculations in random matrix theory.

Findings

Derived the leading correction to the energy density-density correlation at large energy differences.

Calculated the leading correction to the two-point correlation function at small energy differences.

Addressed potential divergences from unbounded saddle points in the supersymmetry approach.

Abstract

The leading correction to the smoothed connected energy density-density correlation function is obtained for the large energy difference, within the context of the Gaussian Random Matrix Theory. In order to achieve this result, the supersymmetry method is extended beyond the sigma-model, to include small quadratic fluctuations around the saddle point. Special care is taken to avoid the potential divergence arising from the unbounded nature of the saddle point. Also, in the small energy difference regime, the leading correction to a two point correlation function is obtained.