Eigenvalue fluctuations for the crushed ice problem

Source abstract

We study the eigenvalues of the negative Dirichlet Laplacian $-Δ$ in domain $D_{ε,N}$ obtained by removing $N$ Euclidean balls of radius $ε$ from a bounded open set $D\subseteq\mathbb R^d$ ($d\ge2$). It is known that, for the balls centered at points drawn independently from a distribution with probability density $ρ$ and for $N\to\infty$ and $ε\downarrow0$ so that the aggregate Newtonian capacity of the removed balls tends to some $κ>0$, the eigenvalues of $-Δ$ in $D_{ε,N}$ homogenize to those of the Schrödinger operator $-Δ+κρ$ in $D$. Here we prove that, for the eigenvalues that are simple in the limit, a multivariate Central Limit Theorem holds in all $d\ge2$ under centering by expected values. Unlike earlier work in $d=3$ by Figari, Orlandi and Teta and, independently, Ozawa from 1980s that was based on resolvent expansions, we rely on martingale representations and one-ball perturbation asymptotic controlled by methods of harmonic analysis.