Preprint

Inserted: 30 apr 2025
Last Updated: 30 apr 2025

Year: 2025

Abstract:

While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of \(\mathbb{Z}^d\) when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber-Krahn inequality.

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• Faber_Krahn_final.pdf