Asymptotics for a special solution of the thirty fourth Painlevé equation

Abstract

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form with α > −1/2. The factor | det M|2α induces critical eigenvalue behaviour near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N → ∞ such that n2/3(n/N − 1) = O(1) by using the Deift–Zhou steepest descent method for the Riemann–Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|2αe−NV(x). Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution uα of the Painlevé XXXIV equation. This solution is related to a particular solution of the Painlevé II equation, which, however, is different from the usual Hastings–McLeod solution. In this paper we compute the asymptotic behaviour of uα(s) as s → ±∞. We conjecture that this asymptotics characterizes uα and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painlevé XXXIV equation which includes uα. We identify this family as the family of tronquée solutions of the thirty fourth Painlevé equation.