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

Abstract

We study the Fredholm determinant of an integrable operator acting on the interval (0, s) whose kernel is constructed out of the Ψ-function associated with a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order k in a certain scaling regime. Using the Riemann-Hilbert method, we obtain the large s asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to a coupled Painlevé III system.

Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

“…The proof is similar to [40] and [63, Lemma 1]. The key point is that the entries in the jump matrices J i in (3.12) satisfy the following conjugate relations (J 1 ) 11 = (J 1 ) 22 and (J 2 ) 12 = (J 4 ) 12 . Proof.…”

“…We aim to find analogous expressions of the Tracy-Widom formula for these determinants and evaluate their large gap asymptotics. We also plan to study the P 3 kernel at the hard edge with pole singularities in the potential in a forthcoming publication [22].…”

Gaussian Unitary Ensembles with Pole Singularities Near the Soft Edge and a System of Coupled Painlevé XXXIV Equations

“…Particularly, the new limiting kernel provides a description of the transition between the classical Airy kernel and the Bessel kernel. The results in [29,30] were further extended by Atkin et al in [3], where they considered the case that a fairly general class of potentials perturbed by a pole of order k ∈ N. A hierarchy of higher order analogues to the Painlevé III equation was used to describe the double scaling limits of the partition function and the correlation kernel; see also our recent work [12] on the properties of the Fredholm determinant associated with this family of limiting kernels (known as the gap probability). Other problems related to the singularly perturbed ensembles include the field of spin glasses [1], eigenvalues of Wigner-Smith time-delay matrix in the context of quantum transport and electrical characteristics of chaotic cavities [7,23,27], and the bosonic replica field theories [25].…”

“…Quite recently, the coupled Painlevé systems have appeared frequently in the literature of random matrix theory. For example, in the study of Fredholm determinants associated with the Painlevé II or III kernels, the Tracy-Widom type formulas are given in terms of explicit integrals involving a solution to the coupled Painlevé II [28] or the Painlevé III system [12]. Moreover, the coupled Painlevé II and V systems have also been related to the generating function for the Airy point process and the Bessel point process in [11] and [9], respectively.…”

Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems

“…Expression (28) follows directly from (22) and (27). To derive the coupled Painlevé IV system, we establish four linear equations in the variables and , = 1, 2.…”