An Asymptotic Faber-Krahn Inequality for the Combinatorial Laplacian on Z²

Original · EN

The Faber-Krahn inequality states that among all open domains with a fixed volume in Rⁿ, 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 Z². 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 Gₙ is a sequence of minimizing subgraphs where each Gₙ has n vertices, then after translation the measure of the symmetric difference of Gₙ* and the unit disk converges to 0.