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arXiv · 1810.00741

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

A. B. J. Kuijlaars·L. D. Molag
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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.

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A. B. J. Kuijlaars, L. D. Molag. 2018-10-01. The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$. https://doi.org/10.1088/1361-6544%2Fab247c

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