On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank one case

TL;DR

This paper analyzes the limiting distribution of the largest eigenvalue in a Hermitian matrix model with a rank-one external source, revealing universal and potential-dependent transitional phenomena.

Contribution

It provides a comprehensive computation of the largest eigenvalue distribution for rank-one external sources with general analytic potentials, including non-convex cases.

Findings

Universal transitional behavior for convex potentials.

Potential-dependent transition phenomena for non-convex potentials.

Extension to higher rank external sources in subsequent work.

Abstract

Consider a Hermitian matrix model under an external potential with spiked external source. When the external source is of rank one, we compute the limiting distribution of the largest eigenvalue for general, regular, analytic potential for all values of the external source. There is a transitional phenomenon, which is universal for convex potentials. However, for non-convex potentials, new types of transition may occur. The higher rank external source is analyzed in the subsequent paper.