Spectral statistics for product matrix ensembles of Hermite type with external source

TL;DR

This paper studies the spectral properties of a Hermite-type product matrix ensemble with an external source, revealing phase transitions and deriving new correlation kernels, including Pearcey-type kernels, for eigenvalues on the real line.

Contribution

It introduces a new spectral analysis of Hermite-type product matrices with external sources, including phase transition phenomena and new kernel formulas.

Findings

Eigenvalues form a determinantal point process

Existence of a phase transition at a critical source value

Derivation of Pearcey-type kernels in the critical case

Abstract

We continue investigating spectral properties of a Hermitised random matrix product, which, contrary to previous product ensembles, allows for eigenvalues on the full real line. When a GUE matrix with an external source is involved, we prove that the eigenvalues of the product form a determinantal point process and derive a double integral representation for correlation kernel. As the source changes, we observe a critical value and establish the existence of a phase transition for scaled eigenvalues at the origin. Particularly in the critical case, we obtain a new family of Pearcey-type kernels.