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
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献