Replica Limit of the Toda Lattice Equation

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 Painlevé 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.

4 pages, latex. One reference added. Discussion of GUE now in the main text. Note added. Version to appear in Phys. Rev. Lett