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We get asymptotics for the covering time of inner subsets of a large domain N*D by the confined walk with typical size N. These asymptotics only depend on the set and the first eigenfunction of the Laplace-Beltrami operator on D.

We consider a large domain intersected with random interlacements at low intensities and prove a threshhold under which the capacity of this set becomes negligible. We get a similar theorem for the range up to short times of the random walk conditioned to stay in the domain.

We study properties of the first eigenvector ϕ_N of the Laplace operator on a large discrete set D_N, with typical size N. We use probabilitistic arguments to show its regularity and bounds close to the boundary, also proving the convergence of ϕ_N in the sup norm towards the solution of the Dirichlet eigenvalue problem on D_N/N.

We prove that a random walk conditioned to stay in a large domain of $\ZZ^d$ can be coupled with tilted random interlacements on macroscopic subsets. The tilt is given by the first eigenvector of the Laplacian on the domain, which acts as a drift.