The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ 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 $\theta$ 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 $\theta =\frac{1}{2}$, 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 $\theta =\frac{1}{r}$, 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 $(r+1)\times (r+1)$ 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.

65 pages

References in corpus (5)

• Energy correlations for a random matrix model of disordered bosons
• Universality conjecture and results for a model of several coupled positive-definite matrices
• Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• Large n limit for the product of two coupled random matrices

Cited by in corpus (3)

• Matrix-valued orthogonal polynomials related to hexagon tilings
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble