# Large gap asymptotics at the hard edge for Muttalib-Borodin ensembles

CATW01 - The complex analysis toolbox: new techniques and perspectives

I will present joint work with Christophe Charlier and Julian Mauersberger.
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 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 $s$.

This talk is part of the Isaac Newton Institute Seminar Series series.