An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z^2

TL;DR

This paper demonstrates that as the number of vertices grows, the shape of subgraphs minimizing the first eigenvalue of the combinatorial Laplacian in Z^2 approaches a disk, establishing an asymptotic Faber-Krahn inequality.

Contribution

It introduces an asymptotic analogue of the Faber-Krahn inequality for the combinatorial Laplacian on Z^2, showing minimizers become disk-shaped as size increases.

Findings

Minimizing subgraphs converge to disk shape

Asymptotic shape optimization for discrete Laplacian

Establishment of a discrete Faber-Krahn inequality

Abstract

The Faber-Krahn inequality states that among all open domains with a fixed volume in R^n, the ball minimizes the first Dirichlet eigenvalue of the Laplacian. We study an asymptotic discrete analogue of this for the combinatorial Dirichlet Laplacian acting on induced subgraphs of Z^2. Namely, an induced subgraph G with n vertices is called a minimizing subgraph if it minimizes the first eigenvalue of the combinatorial Dirichlet Laplacian among all induced subgraphs with n vertices. Consider an induced subgraph G and take the interior of the union of closed squares of area 1 about each point of G. Let G* denote this domain scaled down to have area 1. Our main theorem states that if {G_n} is a sequence of minimizing subgraphs where each G_n has n vertices, then after translation the measure of the symmetric difference of G_n* and the unit disk converges to 0.