Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib-Borodin Ensembles

Abstract

We consider the limiting process that arises at the hard edge of Muttalib-Borodin ensembles. This point process depends on $\theta >0$ and has a kernel built out of Wright's generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form $\begin{array}{c}P\left(\text{gap}\text{on}[0,s]\right)=Cexp\left(-,a,s^{2\rho },+,b,s^{\rho },+,c,ln,s\right)(1+o\left(1\right))\text{as}s\to +\infty ,\end{array}$ where the constants $\rho$ , a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann-Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in $\theta$ . When $\theta$ is rational, we find that C can be expressed in terms of Barnes' G-function. We also show that the asymptotic formula can be extended to all orders in s.