The Parabolic Anderson Model

König, Wolfgang

doi:10.1007/978-3-319-33596-4

Cited by 94 publications

(60 citation statements)

References 111 publications

(269 reference statements)

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“…Previously, there has been a huge amount of work devoted to the study of various localization phenomenon when the potential distribution exhibits some tail behavior ranging from heavy tail to doubly-exponential tail. See [34] for an almost up-to-date review on this subject, also known as the parabolic Anderson model. See also [9] for a review on random walk among mobile/immobile random traps.…”

Section: Background and Related Resultsmentioning

confidence: 99%

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

Ding

2020

Commun. Math. Phys.

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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.

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Section: Background and Related Resultsmentioning

confidence: 99%

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

Ding

2020

Commun. Math. Phys.

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“…The generator of the killed random walk can be formally written as the random Schrödinger operator − 1 2d ∆+∞·½ O , where ∆ is the discrete Laplacian. For this type of operators, various localization phenomena have been predicted and some of them have been rigorously proved; see e.g., [1,12]. In particular, the corresponding parabolic problem in our setting is the discrete time initial-boundary value problem      u(n + 1, x) − u(n, x) = 1 2d ∆u(n, x), (n, x) ∈ Z + × (Z d \ O), u(n, x) = 0, (n, x) ∈ Z + × O, u(0, x) = ½ {0} (x), (1.4) and the probability P(S t = x, τ > n) represents its unique bounded solution.…”

Section: Model and Main Resultsmentioning

confidence: 99%

“…The first step to tackle this problem is to understand how the environment looks like in the localization region, which is an interesting problem itself. These two problems are listed as the main questions in [12,Section 1.3]. The first main result in this paper is about the behavior of environment in the localization region.…”

Section: Model and Main Resultsmentioning

confidence: 99%

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

Ding

Fukushima

Sun

et al. 2019

Probab. Theory Relat. Fields

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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.

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“…This began with the seminal paper [6] and is by now well-understood, see [7,10,13] for surveys. In the i.i.d.…”

Section: Localisation In the Pammentioning

confidence: 99%

Delocalising the parabolic Anderson model through partial duplication of the potential

Muirhead

Pymar

Sidorova

2017

Probab. Theory Relat. Fields

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The parabolic Anderson model on Z d with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites.

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The Parabolic Anderson Model

Cited by 94 publications

References 111 publications

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

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

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

Delocalising the parabolic Anderson model through partial duplication of the potential

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