Gap probabilities for the Bures-Hall Ensemble and the Cauchy-Laguerre Two-Matrix Model

TL;DR

This paper studies the probability of spectral gaps in the Bures-Hall ensemble, connecting it to a two-matrix model and developing new polynomial systems and differential equations for analysis.

Contribution

It introduces new results on Cauchy bi-orthogonal polynomials, including a Christoffel-Darboux formula and integrable differential equations relevant to the Bures-Hall spectrum.

Findings

Derived new Christoffel-Darboux formula for Cauchy bi-orthogonal polynomials

Established a system of nonlinear differential equations for spectral gap probabilities

Connected spectral gap analysis to integrable Lax equations

Abstract

The Bures metric and the associated Bures-Hall measure is arguably the best choice for studying the spectrum of the quantum mechanical density matrix with no apriori knowledge of the system. We investigate the probability of a gap in the spectrum of this model, either at the bottom $ [0,s) $ or at the top $ (s,1] $, utilising the connection of this Pfaffian point-process with the allied problem in the determinantal point-process of the two-dimensional Cauchy-Laguerre bi-orthogonal polynomial system, now deformed with two variables $s,t$. To this end we develop new general results about Cauchy bi-orthogonal polynomial system for a more general class of weights than the Laguerre densities: in particular a new Christoffel-Darboux formula, reproducing kernels and differential equations for the polynomials and their associated functions. This system is most simply expressed as rank-3 matrix variables and possesses an associated cubic bilinear form. Furthermore under specialisation to truncated Laguerre type densities for the weight, of direct relevance to the Cauchy-Laguerre system, we construct a closed system of constrained, nonlinear differential equations in two deformation variables $s,t$, and observe that the recurrence, spectral and deformation derivative structures form a compatible and integrable triplet of Lax equations.