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Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

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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 2 I 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 2 I 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 2 I equation pops up in the n −2/7-term of the asymptotics.

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  1. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001, Leuven, Belgium

    T. Claeys & M. Vanlessen

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  1. T. Claeys
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  2. M. Vanlessen
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Correspondence to T. Claeys.

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Communicated by B. Simon.

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Claeys, T., Vanlessen, M. Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models. Commun. Math. Phys. 273, 499–532 (2007). https://doi.org/10.1007/s00220-007-0256-9

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