Asymptotics for a special solution of the thirty fourth Painleve equation

TL;DR

This paper analyzes the asymptotic behavior of a special solution to the Painleve XXXIV equation, which arises in the double scaling limit of certain random matrix ensembles, revealing new insights into its asymptotics and relation to Painleve II.

Contribution

It computes the asymptotic behavior of a special Painleve XXXIV solution and links it to the tronquee solutions, advancing understanding of critical eigenvalue behavior in random matrices.

Findings

Asymptotic behavior of u_{α}(s) as s → ±∞ is determined.

The special solution u_{α} is identified as a tronquee solution of Painleve XXXIV.

Supports conjecture that the asymptotics characterize u_{α}.

Abstract

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \Tr V(M)} dM with \alpha > -1/2. The factor |\det M|^{2\alpha} induces critical eigenvalue behavior 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 \to \infty such that n^{2/3}(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|^{2\alpha} e^{-NV(x)}. Our main attention was on the construction of a local parametrix near the origin by means of the \psi-functions associated with a distinguished solution u_{\alpha} of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behavior of u_{\alpha}(s) as s \to \pm \infty. We conjecture that this asymptotics characterizes u_{\alpha} and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u_{\alpha}. We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.