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

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.

51 pages

References in corpus (2)

• Energy correlations for a random matrix model of disordered bosons
• Universality conjecture and results for a model of several coupled positive-definite matrices

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