On Expanding Neighborhoods of Local Universality of Gaussian Unitary Ensembles

Abstract

The classical universality theorem states that the Christoffel–Darboux kernel of the Hermite polynomials scaled by a factor of $1/\sqrt n$ tends to the sine kernel in local variables $\tilde x,\tilde y$ in a neighborhood of a point $x^*\in(-\sqrt 2,\sqrt 2)$). This classical result is well known for $\tilde x,\tilde y\in{K}\Subset\mathbb{R}$. In this paper, we show that this classical result remains valid for expanding compact sets K = K(n). An interesting phenomenon of admissible dependence of the expansion rate of compact sets K(n) on x* is established. For $x^*\in(-\sqrt 2,\sqrt 2)\backslash\left\{0\right\}$) and for x* = 0, there are different growth regimes of compact sets K(n). A transient regime is found.