Universality of correlation functions of hermitian random matrices in an external field

TL;DR

This paper demonstrates that the short-distance correlation functions of a broad class of hermitian random matrix models with external fields exhibit universal behavior in the large N limit, extending known universality results.

Contribution

It extends the universality of correlation functions to matrix models with external sources, using a matrix integral formulation and asymptotic analysis of a kernel.

Findings

Short-distance correlation functions are universal in the large N limit.

Finite N determinant formulas reduce the problem to analyzing a single kernel.

Multi-matrix generalizations are discussed.

Abstract

The behavior of correlation functions is studied in a class of matrix models characterized by a measure $\exp(-S)$ containing a potential term and an external source term: $S=N\tr(V(M)-MA)$. In the large $N$ limit, the short-distance behavior is found to be identical to the one obtained in previously studied matrix models, thus extending the universality of the level-spacing distribution. The calculation of correlation functions involves (finite $N$) determinant formulae, reducing the problem to the large $N$ asymptotic analysis of a single kernel $K$. This is performed by an appropriate matrix integral formulation of $K$. Multi-matrix generalizations of these results are discussed.