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

We consider the limiting process that arises at the hard edge of Muttalib-Borodin ensembles. This point process depends on θ > 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 P ( gap on [ 0 , s ] ) = C exp - a s 2 ρ + b s ρ + c ln s ( 1 + o ( 1 ) ) as s → + ∞ , where the constants ρ , 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 θ . When θ 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.