Leslie Molag: A Riemann-Hilbert approach to the Muttalib-Borodin ensemble

Title:

A Riemann-Hilbert approach to the Muttalib-Borodin ensemble

Abstract: 

The Muttalib-Borodin ensemble is a probability density function for n particles on the positive half line, depending on a fixed parameter θ and a potential. It was introduced by Muttalib in 1995 to deal with physical systems that were not accurately described by existing probability density functions arising from random matrix theory. Its definition was generalized by Borodin in 1998 who also proved some interesting results for specific choices of the potential. In our research we are interested in the scaling properties of the ensemble as n tends to infinity, for general potentials. When θ = 1/r for r a natural number we can view the ensemble as a type II multiple orthogonal polynomial ensemble. There is an (r+1)x(r+1) Riemann-Hilbert problem associated to such an orthogonal polynomial ensemble. For simplicity we will mainly be focusing on the θ = ½ case. It is my goal to explain to the audience what ingredients are needed for solving such a Riemann-Hilbert problem, without getting too technical. It is to be noted that this research is not finished yet, I will present some partial results and discuss what results I expect to find eventually.