Integrable Systems and Random Matrices
Baik, Jinho, Kriecherbauer, Thomas, Luen-Chau Li, K. T-R McLaughlin, Tomei, Carlos · Contemporary mathematics - American Mathematical Society · 2008
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.