On Wright's generalized Bessel kernel

Abstract

In this paper, we consider the Wright's generalized Bessel kernel $K^{(\alpha ,\theta )}(x,y)$ defined by $$\theta x^{\alpha }{\int }_0^1{J}_{\frac{\alpha +1}{\theta },\frac{1}{\theta }}(ux)J_{\alpha +1,\theta }((uy{)}^{\theta })u^{\alpha }\mathrm{d}u,\alpha >-1,\theta >0,$$ where $$J_{a,b}(x)=\sum _{j=0}^{\infty }\frac{(-x{)}^j}{j!\Gamma (a+bj)},a\in \mathbb{C},b>-1,$$ is Wright's generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to $\theta =1$) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles. We show that, if $\theta$ is rational, i.e., $\theta =\frac{m}{n}$ with $m,n\in \mathbb{N}$, $gcd(m,n)=1$, and $\alpha >m-1-\frac{m}{n}$, the Wright's generalized Bessel kernel is integrable in the sense of Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this kernel over the union of several scaled intervals, which can also be interpreted as the gap probability (the probability of finding no particles) on these intervals. The integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding Fredholm determinant as well as a Hamiltonian interpretation. As a consequence, we are able to represent the gap probability over a single interval $(0,s)$ in terms of a solution of a system of nonlinear ordinary differential equations.

25 pages, 1 figure. Title changed, size reduced, added numerics of gap probabilities and its small s asymptotics for general parameters, to appear in Physica D: Nonlinear Phenomena

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• Analytical properties and applications of the Wright function
• Energy correlations for a random matrix model of disordered bosons
• Differential equations for singular values of products of Ginibre random matrices
• Raney distributions and random matrix theory
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Cited by in corpus (8)

• The local universality of Muttalib-Borodin biorthogonal ensembles with parameter $\theta =\frac{1}{2}$
• The local universality of Muttalib-Borodin ensembles when the parameter $\theta$ is the reciprocal of an integer
• Integrable structure of products of finite complex Ginibre random matrices
• Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential
• Global rigidity and exponential moments for soft and hard edge point processes
• Gap probability for products of random matrices in the critical regime
• Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
• On the computation of density and two-point correlation functions of a class of random matrix ensembles