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

Abstract

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.