Universality in unitary random matrix ensembles when the soft edge meets the hard edge
(2008) Contemporary Mathematics — Vol. 458, n° Integrable Systems and Random Matrices: in honor of Percy Deift, p. 265-280 (2008)
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- 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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Claeys, T., & Kuijlaars, A. (2008). Universality in unitary random matrix ensembles when the soft edge meets the hard edge. Contemporary Mathematics, 458(Integrable Systems and Random Matrices: in honor of Percy Deift), 265-280. https://doi.org/10.1090/conm/458 (Original work published 2008)