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Universality in unitary random matrix ensembles when the soft edge meets the hard edge

Tom Claeys·Arno B. J. Kuijlaars
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Abstract

Unitary random matrix ensembles Z_{n,N}^{-1} (\det M)^alpha exp(-N Tr V(M)) dM defined on positive definite matrices M, where alpha > -1 and V is real analytic, have a hard edge at 0. The equilibrium measure associated with V typically vanishes like a square root at soft edges of the spectrum. For the case that the equilibrium measure vanishes like a square root at 0, we determine the scaling limits of the eigenvalue correlation kernel near 0 in the limit when n, N tend to infinity such that n/N - 1 = O(n^{-2/3}). For each value of alpha > -1 we find a one-parameter family of limiting kernels that we describe in terms of the Hastings-McLeod solution of the Painleve II equation with parameter alpha + 1/2.

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Tom Claeys, Arno B. J. Kuijlaars. 2007-06-22. Universality in unitary random matrix ensembles when the soft edge meets the hard edge. https://arxiv.org/abs/math-ph/0701003

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