The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

TL;DR

This paper analyzes the local universality of the Muttalib-Borodin biorthogonal ensemble for parameter rac12, deriving the large n behavior of the correlation kernel using Riemann-Hilbert techniques and a novel iterative matching method.

Contribution

It establishes the local universality for rac12 in the Muttalib-Borodin ensemble and introduces a new iterative technique for matching parametrices in Riemann-Hilbert problems.

Findings

Derived the large n correlation kernel behavior for rac12

Developed a new iterative matching technique for Riemann-Hilbert problems

Extended universality results to a broader class of external fields

Abstract

The Muttalib-Borodin biorthogonal ensemble is a probability density function for $n$ particles on the positive real line that depends on a parameter $\theta$ and an external field $V$. For $\theta=\frac{1}{2}$ we find the large $n$ behavior of the associated correlation kernel with only few restrictions on $V$. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a $3\times 3$ Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.