The local universality of Muttalib-Borodin ensembles

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 $w$. The Model was introduced by Muttalib in 1995 to better model systems of disordered conductors in the metallic regime. In 1999 Borodin found a hard edge scaling limit at the origin for several specific choices of the weight $w$. In an article with Arno Kuijlaars, we proved that this limit is in fact universal for $\theta=1/2$, i.e., the limit holds for a large class of weights. Very recently, I generalized the techniques of this article to prove that the limit is universal when $\theta=1/r$, 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, and it is matched with the global parametrix with the help of a double matching, a technique that was recently introduced.