Universality in Unitary Random Matrix Ensembles When the Soft Edge Meets the Hard Edge

Unitary random matrix ensembles Z n,N (detM) α exp(−N TrV (M)) dM defined on positive definite matrices M , where α > −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 → ∞ such that n/N−1 = O(n−2/3). For each value of α > −1 we find a one-parameter family of limiting kernels that we describe in terms of the Hastings-McLeod solution of the Painlevé II equation with parameter α+ 1/2.