Deviations for the capacity of the range of a random walk

Asselah, Amine; Schapira, Bruno

doi:10.1214/20-ejp560

Cited by 9 publications

(20 citation statements)

References 17 publications

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“…Proof. To prove the theorem we use the same reasoning as in the proof of Theorem 1.1 but we use results from [1] instead of results from [7]. More precisely, instead of [7, Lemma…”

Section: Condition (Ii)mentioning

confidence: 99%

“…where the random variables  (1) and  (2) are independent, and have the same law as  and  , respectively. Moreover, the random variable ( , ) has the same law as | ∩  |.…”

Section: Fclt For the Process {| |} ≥0mentioning

confidence: 99%

“…The Markov property implies that the random variables  (1) and  (2) are independent and that the law of  (2) is equal to the law of  . By symmetry,  (1) has the same law as  . Evidently, the random variable | (1) ∩  (2) | is equal in law to | ∩  |.…”

Section: Fclt For the Process {| |} ≥0mentioning

confidence: 99%

“…Studies on the long-time behavior of { } ≥0 were initiated by Jain and Orey in [10] where they obtained a version of the strong law of large numbers for any transient random walk. Asselah, Schapira and Sousi [1] proved a central limit theorem for { } ≥0 of a simple random walk in ≥ 6. Versions of the law of large numbers and central limit theorem in = 4 were proved by the same authors in [2], see also [6].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Functional CLT for the range of stable random walks

Cygan¹,

Sandrić²,

Šebek³

2019

Preprint

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“…Proof. To prove the theorem we use the same reasoning as in the proof of Theorem 1.1 but we use results from [1] instead of results from [7]. More precisely, instead of [7, Lemma…”

Section: Condition (Ii)mentioning

confidence: 99%

Section: Fclt For the Process {| |} ≥0mentioning

confidence: 99%

Section: Fclt For the Process {| |} ≥0mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Functional CLT for the range of stable random walks

Cygan¹,

Sandrić²,

Šebek³

2019

Preprint

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“…The behaviour of the capacity of the range of a simple random walk in high dimension has been thoroughly studied in [4,5,27,40]. For the purposes of this work, we only need a law of large numbers and a large deviations results.…”

Section: Estimates On the Capacity And The Intersections Of Random Walksmentioning

confidence: 99%

The chemical distance in random interlacements in the low-intensity regime

Hernández-Torres¹,

Procaccia²,

Rosenthal³

2021

Preprint

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In Z d with d ≥ 5, we consider the time constant ρ u associated to the chemical distance in random interlacements at low intensity u ≪ 1. We prove an upper bound of order u −1/2 and a lower bound of order u −1/2+ε . The upper bound agrees with the conjectured scale in which u 1/2 ρ u converges to a constant multiple of the Euclidean norm, as u → 0. Along the proof, we obtain a local lower bound on the chemical distance between the boundaries of two concentric boxes, which might be of independent interest. For both upper and lower bounds, the paper employs probabilistic bounds holding as u → 0; these bounds can be relevant in future studies of the low-intensity geometry.

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Large Deviations for Intersections of Random Walks

Asselah

Schapira

2022

Comm Pure Appl Math

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We prove a large deviations principle for the number of intersections of two independent infinite-time ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finite-time horizon. The proof builds on their result (which was adapted in the discrete setting by Phetpradap [12]), and combines it with a series of tools that were developed in recent works of the authors [2,3,5]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order .

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Deviations for the capacity of the range of a random walk

Cited by 9 publications

References 17 publications

Functional CLT for the range of stable random walks

Functional CLT for the range of stable random walks

The chemical distance in random interlacements in the low-intensity regime

Large Deviations for Intersections of Random Walks

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