Random hermitian matrices in an external field

TL;DR

This paper introduces a model of random Hermitian matrices with a U(N)-invariant potential and external source, providing exact finite-N correlation functions that facilitate asymptotic analysis.

Contribution

It generalizes determinant formulas for correlation functions in Hermitian matrix models with external sources, offering compact expressions valid at finite N.

Findings

Exact finite-N correlation functions derived

Potential for asymptotic analysis demonstrated

Generalized determinant formulas established

Abstract

In this article, a model of random hermitian matrices is considered, in which the measure $\exp(-S)$ contains a general U(N)-invariant potential and an external source term: $S=N\tr(V(M)+MA)$. The generalization of known determinant formulae leads to compact expressions for the correlation functions of the energy levels. These expressions, exact at finite $N$, are potentially useful for asymptotic analysis.