The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

Abstract

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the kth largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the kth and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.

Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

“…For ∈ (0, 1], F (s; ) = P(ζ ( ) max < s) is the probability distribution of the largest particle ζ ( ) max in the associated thinned process, which is obtained from the original process by removing each of the particles independently with probability 1 − , see e.g. [8,10,11,12]. In Appendix A, we confirm using standard methods that the kernels (1.1) indeed define a point process for any choice of n, τ 1 , .…”

“…12 (−s) = −2iΨ 1,21 (−s), for n even,(2.27)is real for s ∈ R, 0 < ρ ≤ 1, and it solves the Painlevé II hierarchy equation(1.4).…”

The generating function for the Bessel point process and a system of coupled Painlevé V equations

“…The goal of this paper is to express F ( x, s) explicitly in terms of k functions which satisfy a system of k coupled Painlevé V equations. Analogous generating functions for the Airy point process have been recently studied in [6] (for a general k ∈ N >0 ), and in [26] (for the case k = 2 with an extra root-type singularity). In both cases, the authors expressed it in terms of a system of coupled Painlevé II equations.…”

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

“…It is worthwhile to note that the relations between Fredholm determinants and other coupled Painlevé systems have also been established in recent studies, which generalize the classical results of Tracy and Widom [38,39]. More precisely, the coupled Painlevé II systems have been related to the generating function for the Airy point process in [15], to the Fredholm determinants of the Painlevé II and the Painlevé XXXIV kernel in [42]. A coupled Painlevé V system has been related to the generating function for the Bessel point process in [12].…”

“…where c V is a constant depending on the regular part V in the potential. It was shown in [3, Theorem 1.10] that 15) uniformly for u, v and λ in any compact subsets of (0, ∞), where…”