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Asymptotics for a special solution of the thirty fourth Painleve equation
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2009 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 7, 1523-1558 p.Article in journal (Refereed) Published
Abstract [en]

In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z(n,N)(-1) vertical bar det M vertical bar(2 alpha)e(-NTrV(M)) dM with alpha > -1/2. The factor vertical bar det M vertical bar(2 alpha) 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 -> infinity 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 vertical bar x vertical bar(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 behaviour of u(alpha)(s) as s -> +/-infinity. 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.

Place, publisher, year, edition, pages
2009. Vol. 22, no 7, 1523-1558 p.
National Category
URN: urn:nbn:se:uu:diva-128993DOI: 10.1088/0951-7715/22/7/002ISI: 000266916100003OAI: oai:DiVA.org:uu-128993DiVA: diva2:332418
Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2016-05-13Bibliographically approved

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Östensson, Jörgen
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