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Place an obstacle with probability 1−p independently at each vertex of Z d , and run a simple random walk until hitting one of the obstacles. For d ≥ 2 and p strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that for environments with probability tending to one as n → ∞ there exists a unique discrete Euclidean ball of volume d log 1/p n asymptotically such that the following holds: conditioned on survival up to time n we have that at any time t ∈ [ n n, n] (for some n → n→∞ 0) with probability tending to one the simple random walk is in this ball. This work relies on and substantially improves a previous result of the authors on localization in a region of volume poly-logarithmic in n for the same problem.

Place an obstacle with probability 1 − p independently at each vertex of Z d , and run a simple random walk until hitting one of the obstacles. For d ≥ 2 and p strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as n → ∞: conditioned on survival up to time n we have that ever since o(n) steps the simple random walk is localized in a region of volume poly-logarithmic in n with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume t o(1) was derived conditioned on the survival of Brownian motion up to time t.1. Introduction. For d ≥ 2, we consider a random environment where each vertex of Z d is occupied by an obstacle independently with probability 1 − p ∈ (0, 1). Given this random environment, we then consider a discrete time simple random walk (S t ) t∈N started at the origin and killed at the first time τ when it hits an obstacle. In this paper, we study the quenched behavior of the random walk conditioned on survival for a large time, and we prove the following localization result. For convenience of notation, throughout the paper we use P (and E) for the probability measure with respect to the random environment, and use P (and E) for the probability measure with respect to the random walk.

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Place an obstacle with probability 1 − p independently at each vertex of Z d and consider a simple symmetric random walk that is killed upon hitting one of the obstacles. For d ≥ 2 and p strictly above the critical threshold for site percolation, we condition on the environment such that the origin is contained in an infinite connected component free of obstacles. It has previously been shown that with high probability, the random walk conditioned on survival up to time n will be localized in a ball of volume asymptotically d log 1/p n. In this work, we prove that this ball is free of obstacles, and we derive the limiting one-time distributions of the random walk conditioned on survival. Our proof is based on obstacle modifications and estimates on how such modifications affect the probability of the obstacle configurations as well as their associated Dirichlet eigenvalues, which is of independent interest. MSC 2000. Primary: 60K37; Secondary: 60K35.