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

TL;DR

This paper analyzes the asymptotic behavior of a new class of Fredholm determinants related to higher order Airy kernels, revealing their connection to Painlevé II hierarchy solutions and applications in random point processes.

Contribution

It establishes the asymptotics of higher order Airy-kernel determinants using Riemann-Hilbert methods and links them to Painlevé II hierarchy solutions, extending the Tracy-Widom distribution.

Findings

Asymptotics of determinants as x→−∞ for different ρ regimes

Explicit constant term calculation for 0<|ρ|<1

Singular behaviors in asymptotics for |ρ|>1

Abstract

We study the one-parameter family of Fredholm determinants $\det(I-\rho^2\mathcal{K}_{n,x})$, $\rho\in\mathbb{R}$, where $\mathcal{K}_{n,x}$ stands for the integral operator acting on $L^2(x,+\infty)$ 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 $n$-th member of the Painlev\'{e} II hierarchy. Using the Riemann-Hilbert approach, we establish asymptotics of the determinants and the associated higher order Painlev\'{e} II transcendents as $x\to -\infty$ for $0<|\rho|<1$ and $|\rho|>1$, respectively. In the case of $0<|\rho|<1$, we are able to calculate the constant term in the asymptotic expansion of the determinants, while for $|\rho|>1$, the relevant asymptotics exhibit singular behaviors. 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.