Phase transitions for products of characteristic polynomials under Dyson Brownian motion

TL;DR

This paper investigates phase transitions in the behavior of averaged products of characteristic polynomials in certain random matrix ensembles under Dyson Brownian motion, revealing Pearcey-type transitions and explicit limiting forms.

Contribution

It introduces explicit functional forms for the scaled limits of these products and characterizes phase transitions caused by full rank perturbations of the source.

Findings

Identifies Pearcey-type phase transitions in Gaussian and Laguerre β-ensembles.

Derives explicit multidimensional integral formulas for the limits.

Characterizes phases via functional forms of scaled limits.

Abstract

We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $\beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised by determining the explicit functional forms of the scaled limits of the averaged products of characteristic polynomials, which are given as certain multidimensional integrals, with dimension equal to the number of products.