Replica Limit of the Toda Lattice Equation

TL;DR

This paper derives exact expressions for the resolvent of the chiral Unitary Ensemble using the replica limit of the Toda lattice equation, connecting supersymmetric and replica methods in random matrix theory.

Contribution

It provides a novel analysis of the replica limit of the Toda lattice equation, elucidating the role of supersymmetry in random matrix resolvent calculations.

Findings

Exact resolvent expressions for chiral Unitary Ensemble

Connection between supersymmetric and replica partition functions

Explanation of compact and noncompact integrals in replica limit

Abstract

In a recent breakthrough Kanzieper showed that it is possible to obtain exact nonperturbative Random Matrix results from the replica limit of the corresponding Painlev\'e equation. In this article we analyze the replica limit of the Toda lattice equation and obtain exact expressions for the resolvent of the chiral Unitary Ensemble both in the quenched limit and in the presence of additional massive flavors. This derivation explains in a natural way the appearance of both compact and noncompact integrals, the hallmark of the supersymmetric method, in the replica limit of the expression for the resolvent. We also show that the supersymmetric partition function and the partition function with fermionic replicas are related through the Toda lattice equation.