From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

Fukushima, Ryoki

doi:10.1214/ecp.v14-1497

Cited by 12 publications

(16 citation statements)

References 24 publications

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“…The localization problem in the quenched case was far more challenging: in the Brownian setting it was studied first in [29,32] and its logarithmic asymptotics for the survival probability was first derived in [29] (see also the celebrated monograph [33]). In [12] a simple argument for the quenched asymptotics of the survival probability was given using the Lifshitz tail effect. In the random walk setting, the logarithmic asymptotics of survival probability was computed in [1] which built upon methods developed in the Brownian setting.…”

Section: University Of Pennsylvania and University Of Chicagomentioning

confidence: 99%

Poly-logarithmic localization for random walks among random obstacles

Ding¹,

Xu²

2019

Ann. Probab.

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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 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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Section: University Of Pennsylvania and University Of Chicagomentioning

confidence: 99%

Poly-logarithmic localization for random walks among random obstacles

Ding¹,

Xu²

2019

Ann. Probab.

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“…where λ BM 1 (B(0, 1)) is the principal eigenvalue for the Brownian motion killed on exiting B(0, 1). This result was reproven by Fukushima [9]. For the Brownian motion on some irregular spaces such as the Sierpiński gasket, one also sees a similar phenomenon: rates of the annealed and the quenched asymptotics differ (see [28,29]).…”

Section: 2)mentioning

confidence: 79%

“…To the best of our knowledge the quenched asymptotics for Lévy processes with jumps has not been studied before. In the literature concerning the Brownian motion, one finds two methods: Sznitman's paper [34] estimates u ω (t, x) directly, using his 'enlargement of obstacles' technique for the more difficult upper bound (similar method was used on the Sierpiński gasket in [28]); Fukushima [9] gives elegant arguments for deriving both the upper and the lower quenched bound from respective upper and lower bounds at zero for the integrated density of states of the corresponding Schrödinger operator (being closely related to the annealed upper and lower bounds) -this is done by means of the Dirichlet-Neumann bracketing for the Laplace operator. In our work, we are able to find a counterpart of Fukushima's method for Lévy processes with jumps to obtain the upper bounds.…”

Section: 2)mentioning

confidence: 99%

“…Instead, we estimate u ω (t, x) directly. In this part (similarly as in [9,34]), we require the potential profile W to be bounded and compactly supported. We first prove that Q−almost surely there exist sufficiently large regions where the potential is equal to zero, then we force the process to go to this region and then stay there for a long enough time.…”

Section: The Lower Boundmentioning

confidence: 99%

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The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials

Kaleta

Pietruska-Pałuba

2018

Potential Anal

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We study the quenched long time behaviour of the survival probability up to time t, E x e − t 0 V ω (X s)ds , of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential V ω on R d. Such a function is a probabilistic solution to the parabolic equation involving the nonlocal Schrödinger operator based on the generator of the process (X t) t≥0 with potential V ω. For a large class of processes and potentials of finite range, we determine rate functions η(t) and compute explicitly the positive constants C 1 , C 2 such that

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“…This connection was utilized in [Fuk09b] for deriving relations between asymptotics of .E/ for E # inf . …”

Section: Integrated Density Of Statesmentioning

confidence: 99%

The Parabolic Anderson Model

König

2016

Pathways in Mathematics

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Each "Pathways in Mathematics" book offers a roadmap to a currently well developing mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educational style, i.e., in a way that is accessible for advanced undergraduate and graduate students. It also serves as an introduction to and survey of the field for researchers who want to be quickly informed about the state of the art. The point of departure is typically a bachelor/masters level background, from which the reader is expeditiously guided to the frontiers. This is achieved by focusing on ideas and concepts underlying the development of the subject while keeping technicalities to a minimum. Each volume contains an extensive annotated bibliography as well as a discussion of open problems and future research directions as recommendations for starting new projects More information about this series at

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From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

Cited by 12 publications

References 24 publications

Poly-logarithmic localization for random walks among random obstacles

Poly-logarithmic localization for random walks among random obstacles

The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials

The Parabolic Anderson Model

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