The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer

Autor: Molag, L. D.
Rok vydání: 2020
Předmět:
Zdroj: Nonlinearity 34 (2021) 3485-3565
Druh dokumentu: Working Paper
DOI: 10.1088/1361-6544/abeab6
Popis: 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.
Comment: 65 pages
Databáze: arXiv