A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

Abstract

In this paper, we consider the Muttalib-Borodin ensemble of Laguerre type, a determinantal point process on [0,∞) which depends on the varying weights xαe−nV(x), α>−1, and a parameter θ. For θ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions V as n→∞. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in special cases and conjectured to be universal. Our proof is based on a Deift/Zhou nonlinear steepest descent analysis of two 1×2 vector-valued Riemann-Hilbert problems that characterize the biorthogonal polynomials and the explicit construction of (θ+1)×(θ+1)-dimensional local parametrices near the origin in terms of Meijer G-functions.