The Quantitative Faber-Krahn Inequality for the Combinatorial Laplacian in $\mathbb{Z}^{d}$

TL;DR

The paper tackles the discrete Faber–Krahn problem for the combinatorial Laplacian on subsets of $\mathbb{Z}^d$, where minimizers of the first Dirichlet eigenvalue can exhibit non-rigid, fluctuating behavior. By embedding the discrete problem into the continuum via a Kuhn-based extension and employing a quantitative Faber–Krahn inequality, the authors derive sharp fluctuation bounds between almost-minimizing configurations: the symmetric difference between two near-minimizers scales at most like $N^{1-\frac{1}{2d}}$. The approach hinges on a controlled discrete-to-continuum extension of sets and eigenfunctions, together with discrete rearrangement inequalities and $\Gamma$-convergence results, to compare discrete eigenvalues with their continuum counterparts. The results provide explicit rates for how discrete minimizers approximate continuum balls and extent to almost-minimizers, illuminating the connection between discrete spectral optimization and continuum isoperimetric-type stability with potential applications in lattice-based spectral problems.

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.

Theorems & Definitions (38)

• Proposition 2.1
• proof
• Definition 2.2
• Definition 2.3
• Remark 2.4: Energy scalings
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Remark 2.8
• Definition 2.9