On expanding neighborhoods of local universality of Gaussian unitary ensembles

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 $\widetilde x,\widetilde y$ in a neighborhood of a point $x^*\in(-\sqrt2,\sqrt2)$. This classical result is well known for $\widetilde x,\widetilde y\in K\Subset\mathbb R$. 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 $x^*\in(-\sqrt2,\sqrt2)\setminus\{0\}$ and for $x^*=0$, there are different growth regimes of compact sets $K(n)$. A transient regime is found.