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

Abstract

AbstractWe consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on $$\theta > 0$$ 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 $$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$ 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 $$\rho $$ 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 $$\theta $$ θ . When $$\theta $$ 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.