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

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.

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

References in corpus (2)

• Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
• Universality of a double scaling limit near singular edge points in random matrix models

Cited by in corpus (1)

• The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the "birth of a cut" transition