Asymptotics of the deformed higher order Airy-kernel determinants and applications

Author:

Xia JunORCID,Hao Yi-Fan,Xu Shuai-XiaORCID,Zhang Lun,Zhao Yu-Qiu

Abstract

Abstract We study the one-parameter family of Fredholm determinants det ( I ρ 2 K n , x ) , ρ R , where K n , x stands for the integral operator acting on L 2 ( x , + ) with the higher order Airy kernel. This family of determinants represents a new universal class of distributions which is a higher order analogue of the classical Tracy–Widom distribution. Each of the determinants admits an integral representation in terms of a special real solution to the nth member of the Painlevé II hierarchy. Using the Riemann–Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlevé II transcendents as x for 0 < | ρ | < 1 and | ρ | > 1 , respectively. In the case of 0 < | ρ | < 1 , we are able to calculate the constant term in the asymptotic expansion of the determinants, while for | ρ | > 1 , the relevant asymptotics exhibit singular behaviours. Applications of our results are also discussed, which particularly include asymptotic statistical properties of the counting function for the random point process defined by the higher order Airy kernel.

Funder

National Natural Science Foundation of China

Guangdong Basic and Applied Basic Research Foundation

Publisher

IOP Publishing

Subject

Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics

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