Eigenvalue fluctuations for the crushed ice problem
Abstract
We study the eigenvalues of the negative Dirichlet Laplacian −Δ in domain Dε,N obtained by removing N Euclidean balls of radius ε from a bounded open set D⊆ℝd (d≥2). It is known that, for the balls centered at points drawn independently from a distribution with probability density ρ and for N→∞ and ε↓0 so that the aggregate Newtonian capacity of the removed balls tends to some κ>0, the eigenvalues of −Δ in Dε,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≥2 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.
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BibTeX
@misc{biskup2026eigenvalue,
title = {{Eigenvalue fluctuations for the crushed ice problem}},
author = {Marek Biskup and Ryoki Fukushima},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18013},
eprint = {2610.18013},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18013},
}
RIS
TY - UNPB
TI - Eigenvalue fluctuations for the crushed ice problem
AU - Marek Biskup
AU - Ryoki Fukushima
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18013
UR - https://arxiv.org/abs/2610.18013
AB - 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⊆ℝ^d (d≥2). It is known that, for the balls centered at points drawn independently from a distribution with probability density ρ and for N→∞ and ε↓0 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≥2 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.
ER -