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

Source abstract

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 − x2)−μ}, x ∈ ( − 1, 1) for μ > 0. When μ ∈ (0, 1/2) these are Szegö class weights, and are non-Szegö 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).