Localization for Random Walks Among Random Obstacles in a Single Euclidean Ball

Abstract

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.

Geometry of the random walk range conditioned on survival among Bernoulli obstacles

“…Recently, it has been shown in [7,8] that conditioned on survival up to time n, the random walk stays in an island (determined by the environment) of diameter at most polylogarithmic in n during time [o(n), n]. Furthermore, at any deterministic time t ∈ [o(n), n], the random walk stays with high probability in a ball of radius asymptotically…”

“…Intuitively, the ball where the random walk will be localized should contain very few obstacles, or even no obstacle. It is proved in [8] that the volume proportion of obstacles inside the localizing ball is at most o(1), but it remains open to show that it actually contains no obstacle at all. Our first main result resolves this question.…”

“…This latter result is a kind of path localization that had been known only for d = 1. Then in the second paper [8], it is finally proved that the random walk gets localized in a single island, which is called a "one city theorem" in the parabolic Anderson model literature. Our Theorems 1.1 and 1.3 give a more precise picture of the localization established in [7,8].…”

“…Recently, significant improvements on the localization picture have been made in [7,8] for the discrete space-time model. The first paper [7] improved the bound for the number of islands to (log n) C for some constant C, and also proved that the random walk hits the union of these islands rather quickly and then stays there.…”

“…The rest of the paper is organized as follows. We will first collect some results from [7,8] in Section 2, which we will need later in the proof. Then in Section 3, we introduce some intermediate results and use them to prove Theorems 1.1, 1.3.…”