uu.seUppsala University Publications

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Asymptotics for a special solution of the thirty fourth Painleve equationPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2009 (English)In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 22, no 7, 1523-1558 p.Article in journal (Refereed) Published
##### Description

##### Abstract [en]

##### Place, publisher, year, edition, pages

2009. Vol. 22, no 7, 1523-1558 p.
##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-128993DOI: 10.1088/0951-7715/22/7/002ISI: 000266916100003OAI: oai:DiVA.org:uu-128993DiVA: diva2:332418
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Available from: 2010-08-05 Created: 2010-08-05 Last updated: 2017-12-12Bibliographically approved

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.

doi
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