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

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

2007-01-02

arxiv.org/abs/math-ph/0701003v2

Contemporary Mathematics 458 ( 2008), 265--280

Physics

Mathematical Physics

19 pages, 2 figures, one remark and four references added, typos corrected

Scientific paper

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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