On the structure of Normal Matrix Model

TL;DR

This paper analyzes the normal matrix model, demonstrating its exact solvability, universality of correlations, and developing a new integrable hierarchy framework related to the model.

Contribution

It introduces a holomorphic function representation for correlation functions, proves the model's exact solvability, and constructs the extended-KP(N) hierarchies for integrability.

Findings

Correlation functions expressed via holomorphic functions

Two-point correlation function is universal in the scaling limit

Partition function is a tau function of the extended-KP(N) hierarchy

Abstract

We study the structure of the normal matrix model (NMM). We show that all correlation functions of the model with axially symmetric potentials can be expressed in terms of holomorphic functions of one variable. This observation is used to demonstrate the exact solvability of the model. The two-point correlation function is calculated in the scaling limit by solving the BBGKY chain of equations. The answer is shown to be universal (i.e. potential independent up to a change of the scale). We then develop a two-dimensional free fermion formalism and construct a family of completely integrable hierarchies (which we call the extended-KP(N) hierarchies) of non-linear differential equations. The well-known KP hierarchy is a lower-dimensional reduction of this family. The extended-KP(1) hierarchy contains the (2+1)-dimensional Burgers equations. The partition function of the N*N NMM is the tau function of the extended-KP(N) hierarchy invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane.