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

Given a nonnegative integrable function J on \(\mathbb {R}^n\) ${\mathbb{R}}^n$ , we relate the asymptotic properties of the nonlocal energy functional \(\begin{aligned} \int _{\Omega } \int _{\Omega ^c} J \bigg (\frac{x-y}{t}\bigg ) \ dx dy \end{aligned}\) $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}{\int }_{{\mathrm{\Omega }}^c}J(\frac{x-y}{t})dxdy\end{array}$$ as \(t \rightarrow 0^+\) $t\to 0^{+}$ with the boundary properties of a given domain \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , focusing mainly on domains with “rough” boundaries. Then, we apply these results to the fluctuations of many determinantal point processes, showing (under suitable hypotheses) that their variances measure the Minkowski dimension of \(\partial \Omega \) $\partial \mathrm{\Omega }$ .