Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles
AbstractWe consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on$$\theta > 0$$θ>0and 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(gapon[0,s])=Cexp-as2ρ+bsρ+clns(1+o(1))ass→+∞,where the constants$$\rho $$ρ,a, andbhave been derived explicitly via a differential identity insand the analysis of a Riemann–Hilbert problem. Their method can be used to evaluatec(with more efforts), but does not allow for the evaluation ofC. In this work, we obtain expressions for the constantscandCby employing a differential identity in$$\theta $$θ. When$$\theta $$θis rational, we find thatCcan be expressed in terms of Barnes’G-function. We also show that the asymptotic formula can be extended to all orders ins.
