# Higher order large gap asymptotics at the hard edge for Muttalib--Borodin ensembles

## TL;DR

This paper derives explicit formulas for constants in large gap asymptotics at the hard edge of Muttalib--Borodin ensembles, extending previous results to all orders and expressing constants in terms of special functions.

## Contribution

It provides explicit expressions for the constants in the large gap asymptotics, including the constant C, using a differential identity in the parameter , and extends the asymptotic expansion to all orders.

## Key findings

- Explicit formulas for constants c and C in large gap asymptotics.
- Expression of C in terms of Barnes' G-function for rational .
- Extension of asymptotic expansion to all orders in s.

## 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{equation*} \mathbb{P}(\mbox{gap on } [0,s]) = C \exp \left( -a s^{2\rho} + b s^{\rho} + c \ln s \right) (1 + o(1)) \qquad \mbox{as }s \to + \infty, \end{equation*} 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 $s$.