Asymptotics for a special solution of the thirty fourth Painleve equation

Details Asymptotics for a special solution of the thirty fourth Painleve equation Asymptotics for a special solution of the thirty fourth Painleve equation

2008-11-24

arxiv.org/abs/0811.3847v1

Nonlinearity 22 (2009), 1523--1558

Mathematics

Classical Analysis and ODEs

44 pages, 11 figures

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Asymptotics for a special solution of the thirty fourth Painleve equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Asymptotics for a special solution of the thirty fourth Painleve equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Asymptotics for a special solution of the thirty fourth Painleve equation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-645051