Toda lattice representation for random matrix model with logarithmic confinement

TL;DR

This paper develops a replica field theory for a specific random matrix model with logarithmic confinement, deriving exact partition functions and Toda lattice equations, advancing theoretical understanding of such models.

Contribution

It introduces a new replica field theory for the logarithmic confinement random matrix model and derives exact partition functions and Toda lattice equations.

Findings

Exact replica partition function for any matrix size N.

Derived Toda lattice equations for the model.

Representation of the partition function via generalized IZ integral.

Abstract

We construct a replica field theory for a random matrix model with logarithmic confinement [K.A.Muttalib et.al., Phys. Rev. Lett. 71, 471 (1993)]. The corresponding replica partition function is calculated exactly for any size of matrix $N$. We make a color-flavor transformation of the original model and find corresponding Toda lattice equations for the replica partition function in both formulations. The replica partition function in the flavor space is defined by generalized Itzikson-Zuber (IZ) integral over homogeneous factor space of pseudo-unitary supergroups $SU(n\mid M,M)/SU(n\mid M-N,M)$ (Stiefel manifold) with $M\to\infty$, which is evaluated and represented in a compact form.