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Title: Universality of a double scaling limit near singular edge points in random matrix models
Authors: Claeys, Tom ×
Vanlessen, Maarten #
Issue Date: Jul-2007
Publisher: Springer-Verlag Heidelberg
Series Title: Communications in Mathematical Physics vol:273 issue:2 pages:499-532
Abstract: We consider unitary random matrix ensembles $$Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM$$ on the space of Hermitian n × n matrices M, where the confining potential V s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P I 2 equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P I 2 equation.
In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights $$e^{-nV_{s,t}}$$ on $${\mathbb{R}}$$ . The special solution of the P I 2 equation pops up in the n −2/7-term of the asymptotics.
ISSN: 0010-3616
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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