Universality of a double scaling limit near singular edge points in random matrix models
(2007) Communications in mathematical physics — Vol. 273, n° 2, p. 499-532 (2007)
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- Abstract
- We consider unitary random matrix ensembles Z(n,s,t)(-1)e(-ntr) V(s,t(M))dM on the space of Hermitian n x n matrices M, where the confining potential V-s,V-t is such that the limiting mean density of eigenvalues ( as n ->infinity 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 Painleve 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(-nVs,t) on R. The special solution of the P-I(2) equation pops up in the n(-2/7)- term of the asymptotics.
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Claeys, T., & Vanlessen, M. (2007). Universality of a double scaling limit near singular edge points in random matrix models. Communications in mathematical physics, 273(2), 499-532. https://doi.org/10.1007/s00220-007-0256-9 (Original work published 2007)