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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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Original languageEnglish
Pages (from-to)3313–3364
Journal / PublicationAnnales Henri Poincare
Volume20
Issue number10
Online published9 Aug 2019
Publication statusPublished - Oct 2019

Abstract

In this paper, we study the singularly perturbed Gaussian unitary ensembles defined by the measure

(1/Cn)entr V (M;λ,t→) dM

over the space of n × n Hermitian matrices M, where V (x; λ,t) := 2x2 + Σ2mk=1 t(xλ)k with t→ = (t1, t2,...,t2m) ∈ R2m−1 × (0, ∞), in the multiple scaling limit, where λ → 1 together with t → 0 as n → ∞ at appropriate related rates. We obtain the asymptotics of the partition function, which is described explicitly in terms of an integral involving a smooth solution to a new coupled Painlevé system generalizing the Painlevé XXXIV equation. The large n limit of the correlation kernel is also derived, which leads to a new universal class built out of the Ψ-function associated with the coupled Painlevé system.