The First Steklov Eigenvalue on Infinite Subgraphs of Integer Lattices

Abstract

In the context of transient graphs, we study the first Steklov eigenvalue σ0(Ω) of an infinite subgraph with finite boundary (Ω, B) of the integer lattice Zn. We focus in this paper on finding lower bounds using the technique of discretization of smooth compact Riemannian manifolds with cylindrical boundary. These bounds essentially depend on the discretization of the sphere Sn ⊂ Rn+1 with two identical boundaries’ isometrics to {1} × Sn−1 through quasi-isometries. As a consequence, if n ≥ 4 and the boundary B considered as a finite subset of Zd where 1 ≤ d ≤ n − 3, we show that σ0(Ω)|B| 1/n−1 tends to infinity as the cardinal of B tends to infinity. Moreover, if n ≥ 3 and the boundary B is a sphere, we prove that the first Steklov eigenvalue tends to zero proportionally to 1/|B| 1/n−1 as the radius tends to infinity and that σ0(Ω)|B| 1/n−1 is bounded. 2010 Mathematics Subject Classification. 39A12, 05C63, 47B25, 05C12, 05C50.