Fox H-kernel and θ-deformation of the Cauchy Two-Matrix Model and Bures Ensemble

Abstract

A θ-deformation of the Laguerre weighted Cauchy two-matrix model, and the Bures ensemble, is introduced. Such a deformation is familiar from the Muttalib-Borodin ensemble. The θ-deformed Cauchy-Laguerre two-matrix model is a two-component determinantal point process. It is shown that the correlation kernel, and its hard edge scaled limit, can be written as the Fox H-functions, generalising the Meijer G-function class known from the study of the case θ = 1. In the θ = 1 case, it is shown Laguerre-Bures ensemble is related to the Laguerre-Cauchy two-matrix model, notwithstanding the Bures ensemble corresponds to a Pfaffian point process.This carries over to the θ-deformed case, allowing explicit expressions involving Fox H-functions for the correlation kernel, and its hard edge scaling limit, to be obtained.2010 Mathematics Subject Classification. 66B20, 15A15, 33E20.Through the variable transformation x → x/t and y → y/t, we see that the dependence on t can be written as J j,k (t) = t −(1+a+b+θ(j+k−2)) I j,k (a, b; θ) and d dt J j,k (t) = −t −(2+a+b+θ(j+k−2)) (1 + a + b + θ(j + k − 2))I j,k (a, b; θ).

Bures–Hall ensemble: spectral densities and average entropies

“…This is very interesting since the former constitutes a Pfaffian point process, while the latter corresponds to a determinantal point process. Also, very recently, Muttalib-Borodin kind of deformation has been considered in the Cauchy two-matrix model and the unrestricted Bures-Hall ensemble by Forrester and Li [55]. Furthermore, Hu and Li have shown that the partition function of the unrestricted trace Bures-Hall ensemble can be identified as the τ -function of BKP and DKP hierarchies [56].…”

Quantum interpolating ensemble: Average entropies and orthogonal polynomials

“…In the case when θ = 1, the corresponding ensembles have been studied in [33][34][35][36][37]. Among other results in [40], the k-point (1 ≤ k ≤ m) correlation function of the unconstrained ensemble is shown to follow a Pfaffian point process…”

“…. , m. This ensemble has been recently studied in [40] in connection to a θ-deformed Cauchy-Laguerre two-matrix model. In the case when θ = 1, the corresponding ensembles have been studied in [33][34][35][36][37].…”

“…The above kernel functions can be expressed via the following Fox H-functions [40] H q (x) = H 1,1 2,3…”

“…= B q = 1, the Fox H-function reduces to the Meijer G-function [41]. The integral forms of the kernel functions (17a)-(17d) are useful for the computation of mean entropies, whereas several other distinct representations of the kernel functions exist [32,[35][36][37]40]. In particular, we present the following biorthogonal polynomial forms [40] useful for a later discussion…”