Remarks on the thin obstacle problem and constrained Ginibre ensembles

Abstract

We consider the problem of constrained Ginibre ensemble with prescribed portion of eigenvalues on a given curve $\Gamma \subset {\mathbb{R}}^2$ and relate it to a thin obstacle problem. The key step in the proof is the $H^1$ estimate for the logarithmic potential of the equilibrium measure. The coincidence set has two components: one in $\Gamma$ and another one in ${\mathbb{R}}^2\setminus \Gamma$ which are well separated. Our main result here asserts that this obstacle problem is well posed in $H^1({\mathbb{R}}^2)$ which improves previous results in $H_{loc}^1({\mathbb{R}}^2)$.