On Wright’s generalized Bessel kernel

Abstract

In this paper, we consider the Wright's generalized Bessel kernel K (α,θ) (x, y) defined bywhereis Wright's generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to θ = 1) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles. We show that, if θ is rational, i.e., θ = m n with m, n ∈ N, gcd(m, n) = 1, and α > m − 1 − 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.

Large Gap Asymptotics at the Hard Edge for Product Random Matrices and Muttalib–Borodin Ensembles

“…Recently, there has been a lot of interest in Wishart-type products of random matrices [1,2,29,32,33,34] and in Muttalib-Borodin ensembles [9,10,11,13,25,31,35,43,44]. New universal limiting kernels near the hard edge have been discovered in this context, associated to kernels built out of Meijer G-functions [33] and Wright's generalized Bessel functions [10].…”

“…New universal limiting kernels near the hard edge have been discovered in this context, associated to kernels built out of Meijer G-functions [33] and Wright's generalized Bessel functions [10]. The study of the associated Fredholm determinants, which describe the limit distributions of the smallest eigenvalue, was initiated recently in [38] for products of random matrices and in [44] for Muttalib-Borodin ensembles, and remarkable systems of differential equations have been obtained. We contribute to these developments by obtaining large gap asymptotics, and by expressing the Fredholm determinants identically in terms of a 2 × 2 Riemann-Hilbert problem.…”

“…case θ = m n ∈ Q, the Wright's generalized Bessel functions can be expressed as Meijer G-functions as well (see [44]). Furthermore, if θ ∈ N or 1 θ ∈ N, then the limiting kernels K (3) and K (1) are related, as it was shown in [32].…”

“…It is known [33,38] that K (1) is of integrable form with p = r + 1, and that K (3) is of integrable form if θ = a/b ∈ Q with p depending on a and b [44]. Overall, except in the simplest case corresponding to the Bessel kernel, the associated RH problems are of size p × p with p > 2 and it is not clear whether such RH problems can be used to obtain large s asymptotics.…”

Gap Probability at the Hard Edge for Random Matrix Ensembles with Pole Singularities in the Potential

“…It would then be natural to derive the large s asymptotics of det(I − K PIII ) and to ask for its Painlevé type formula, which will be the main results of the present work stated in what follows. We note that similar problems have been addressed for other generalizations of Bessel kernel recently in [16,26,35,45].…”