About the first eigenvector of the simple random walk killed upon exiting a large bounded domain (preprint)

Abstract

In this article, we study a discrete version of a Dirichlet problem in an open bounded set D ⊂ R^d, in dimension d ≥ 2. More precisely, we consider the simple random walk on Z^d, d≥2, killed upon exiting the large (bounded) domain D_N := (ND) ∩ Z^d. We denote by P_N the corresponding transition matrix and we study the properties of its (L2-normalized) principal eigenvector ϕ_N. One of our motivation is that the random walk ``conditioned to stay in DN’’ is a random walk among conductances c_N(x,y) = ϕ_N(x) ϕ_N(y). With probabilistic arguments and under mild assumptions on the domain, we show that ϕ_N varies regularly, with a uniform control inside D_N. We derive several corollaries, among which a uniform convergence of ϕN to the first eigenfunction of the corresponding continuous Dirichlet problem. Our results may not be new, but our proofs use (simple) probabilistic ideas that could be helpful in other contexts.