Nonlocal energy functionals and determinantal point processes on non-smooth domains

Given a nonnegative integrable function $J$ on ${\mathbb{R}}^n$, we relate the asymptotic properties of the nonlocal energy functional \begin{equation*} \int_Ω \int_{Ω^c} J \bigg(\frac{x-y}{t}\bigg) \ dx dy \end{equation*} as $t\to 0^{+}$ with the boundary properties of a given domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Then, we use these asymptotic properties to study the fluctuations of many determinantal point processes, and show that their variances measure the Minkowski dimension of $\mathrm{\partial }\mathrm{\Omega }$.