A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

TL;DR

This paper links the large $n$ behavior of Muttalib-Borodin biorthogonal ensembles to a vector equilibrium problem, especially for rational parameter values, advancing understanding of their asymptotic properties.

Contribution

It establishes an equivalence between the equilibrium problem for these ensembles and a vector equilibrium problem when the parameter $ heta$ is rational.

Findings

Vector equilibrium problem characterizes large $n$ behavior for rational $ heta$

Provides a new framework for analyzing biorthogonal ensembles

Enhances understanding of asymptotic distributions in these ensembles

Abstract

The Muttalib-Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $\theta$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $\theta$ there is an equivalent vector equilibrium problem.