Universality for eigenvalue correlations from the unitary ensemble associated with a family of singular weights

摘要

We study the asymptotic behavior of the eigenvalue correlations for the unitary ensemble associated with a family of singular weights w(x;μ) = exp{-(1 - x~2)~(-μ)}, x ∈ (-1, 1) for μ > 0. When μ ∈ (0, 1/2) these are Szego class weights, and are non-Szego when μ ≥ 1/2. It is proved that the behavior in the bulk of the spectrum is described in terms of the sine kernel, which persists the so-called universality results. While the local behavior at the edge of the spectrum is described in terms of the Airy kernel. A specific scaling of the limit reflects the singular behavior of orthogonal polynomials on [-1, 1], with respect to the weight w(x;μ).