The local universality of Muttalib–Borodin ensembles when the parameter θ is the reciprocal of an integer

Abstract

The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that depends on an external field V. In a recent article, the large n behavior of the associated correlation kernel at the hard edge was found for θ = 1 2 , where only few restrictions are imposed on V. In the current article we generalize the techniques and results of this article to obtain analogous results for θ = 1 r , where r is a positive integer. The approach is to relate the ensemble to a type II multiple orthogonal polynomial ensemble with r weights, which can then be related to an (r + 1) × (r + 1) Riemann–Hilbert problem. The local parametrix around the origin is constructed using Meijer G-functions. We match the local parametrix around the origin with the global parametrix with a double matching, a technique that was recently introduced.

Recursive Characterisations of Random Matrix Ensembles and Associated Combinatorial Objects

“…Thus, it is understood in the present day that the matrix product ensembles reviewed in this section, together with their approximations in terms of Muttalib-Borodin ensembles, belong to a universality class of eigenvalue ensembles that are characterised by having their bulk, soft edge, and (doublesided) hard edge statistics described respectively by the sine kernel, the Airy kernel, and the Meijer G-kernel with various parametrisations. This universality class has consequently garnered great interest, with recent works showing that it contains other matrix product ensembles [122], [216], [205], along with the class of Muttalib-Borodin ensembles with θ > 0 real and general weight of the form w(λ) = λ a e −NV(λ) χ λ>0 [215], [247], [311] (see also [67], [60] for related results on this latter class of Muttalib-Borodin ensembles).…”

Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble

“…Before [15], the special case of (1.6) with f (x) = x θ and w(x) = x α e −x was introduced by Muttalib in [39] from a physical point of view. Hence, the special case of (1.6) with f (x) = x θ is also called the Muttalib-Borodin ensemble, see [29,36,38,20,19] for example. Biorthogonal ensembles and variations thereof are also considered in [5,37,14,17,22,24,9,8,30,31].…”