Density of State in a Complex Random Matrix Theory with External Source

TL;DR

This paper derives a compact integral expression for the density of states in a complex random matrix coupled to an external source, applicable for finite matrix size N.

Contribution

It provides a new integral representation for the density of states in complex random matrices with external sources, extending finite N analysis.

Findings

Derived a compact integral formula for the density of states

Applicable to finite N complex random matrices with external coupling

Facilitates analysis of spectral properties in such systems

Abstract

The density of state for a complex $N\times N$ random matrix coupled to an external deterministic source is considered for a finite N, and a compact expression in an integral representation is obtained.