Given a nonnegative integrable function $J$ on $\mathbb{ R}^n$, we relate the asymptotic properties of the nonlocal energy functional \begin{ equation*} \int_{ \Omega} \int_{ \Omega^c} J \bigg(\frac{ x-y}{ t}\bigg) \ dx dy \end{ equation*} as $t \to 0^+$ with the boundary properties of a given domain $\Omega \subset \mathbb{ R}^n$. 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 $\partial \Omega$.
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