Replica field theories, Painleve transcendents, and exact correlation functions

TL;DR

This paper introduces an exact method for solving nonlinear replica sigma models in random matrix theory, based on a novel relation to Painleve transcendents, avoiding traditional symmetry-breaking assumptions.

Contribution

It presents a new approach linking replica partition functions to Painleve transcendents, enabling exact solutions for various random matrix ensembles without replica symmetry breaking.

Findings

Derived exact solutions for fermionic replicas in GUE and chiral GUE.

Established a general framework applicable to multiple symmetry classes.

Discussed potential applications to other random matrix models.

Abstract

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painleve transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitean random matrices. Further applications are briefly discussed.