  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 $\partial \Omega$.
