Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous, Gérard Ben; Cabezas, Manuel; Fribergh, Alexander

doi:10.1002/cpa.21813

Cited by 9 publications

(8 citation statements)

References 44 publications

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“…A full scaling limit for random walk on a critical Galton-Watson tree conditioned to be infinite was recently proved by Athreya, Löhr, and Winter [ALW17], with a rescaling that, after embedding the tree, exactly corresponds to the one in (1.9). Also recently, Ben-Arous, Cabezas, and Fribergh [BACF16b] proved that the same scaling limit holds in high dimensions (with differentκ andν) for the model where the random walk moves on the trace of the critical BRW in Z d (rather than directly on the tree, as in the current paper).…”

Section: 2supporting

confidence: 71%

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Random walk on barely supercritical branching random walk

Hofstad

Hulshof

Nagel

2019

Probab. Theory Relat. Fields

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Let T be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean µ > 1, conditioned to survive. Let ϕ T be a random embedding of T into Z d according to a simple random walk step distribution. Let T p be percolation on T with parameter p, and let p c = µ −1 be the critical percolation parameter. We consider a random walk (X n ) n≥1 on T p and investigate the behavior of the embedded process ϕ Tp (X n ) as n → ∞ and simultaneously, T p becomes critical, that is, p = p n p c . We show that when we scale time by n/(p n − p c ) 3 and space by (p n − p c )/n, the process (ϕ Tp (X n )) n≥1 converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.MSC 2010. Primary: 60K37, 82C41. Secondary: 60F17, 60K40. Key words and phrases. Branching random walk, random walk indexed by a tree, percolation, scaling limit, supercriticality. arXiv:1804.04396v2 [math.PR] 13 Mar 2019 1.1. Random walk on a randomly embedded random tree. Before we proceed with the main results, let us define the model.Percolation on Galton-Watson trees. Let T be a Galton-Watson tree rooted at with law P, and let ξ be the random number of offspring of the root. Suppose that the tree is supercritical, i.e., µ := E[ξ] > 1. We write ∆ for the maximal number of children that a single individual in the (unpercolated) tree can have, i.e., ∆ := sup{n : P(ξ = n) > 0}.We consider percolation on the edges of the tree and let T p denote the connected component of the root in T , when each edge is deleted with probability 1 − p ∈ (0, 1), independently of every other edge and of T . Then T p is again a Galton-Watson tree, whose distribution we denote by P p . The root of a tree T chosen according to P p now has ξ p offspring, such that, conditioned on {ξ = n}, ξ p is a binomial random variable with parameters n and p.The percolated tree has mean number of offspring pµ and is supercritical if and only if p > p c := 1/µ. In this setting, denote byP p = P p ( · | |T | = ∞) the distribution of the percolated tree, conditioned on non-extinction. Given a tree T rooted at , let (X n ) n≥0 be a simple random walk on T started at . Given v ∈ T , we write P v T for the law of (X n ) n≥0 with X 0 = v, and write P T if v = . Given a realization of a Galton-Watson tree T , we call P T the quenched law of the random walk, and we call P p :=P p × P T the annealed law of the random walk on the tree (writing P v p :Spatial embedding: branching random walk. We embed a Galton-Watson tree T into Z d by means of a branching random walk, which we will define now: Let D denote a non-degenerate probability distribution on Z d . Given a tree T rooted at , set Z( ) = 0 and assign to each vertex v = of T an independent random variable Z(v) with law D. For any v ∈ T there exists a unique path = v 0 , v 1 , . . . , v m = v. The branching random walk embedding ϕ = ϕ T is defined such that ϕ(v) :=...

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Section: 2supporting

confidence: 71%

“…And even in high dimensions, where the picture is perhaps most complete, there are still many big open problems. For instance, it is currently unknown what the scaling limit of random walk on large critical clusters is (although there are good conjectures [HS00,Sla02], on which much progress has been made recently [BACF16a,BACF16b]).…”

Section: Introductionmentioning

confidence: 99%

Random walk on barely supercritical branching random walk

Hofstad

Hulshof

Nagel

2019

Probab. Theory Relat. Fields

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“…Despite immense progress on the understanding of critical and near-critical percolation in two dimensions, it is still not known whether the effective conductivity behaves as a power law near criticality (let alone compute the exponent) in small dimensions. See however [37,36,18], as well as [12,42,13,14] for very fine results in high dimensions.…”

Section: Introductionmentioning

confidence: 99%

Efficient Methods for the Estimation of Homogenized Coefficients

Mourrat

2018

Found Comput Math

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The main goal of this paper is to define and study new methods for the computation of effective coefficients in the homogenization of divergence-form operators with random coefficients. The methods introduced here are proved to have optimal computational complexity, and are shown numerically to display small constant prefactors. In the spirit of multiscale methods, the main idea is to rely on a progressive coarsening of the problem, which we implement via a generalization of the Green-Kubo formula. The technique can be applied more generally to compute the effective diffusivity of any additive functional of a Markov process. In this broader context, we also discuss the alternative possibility of using Monte-Carlo sampling, and show how a simple one-step extrapolation can considerably improve the performance of this alternative method.

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“…The questions studied here are related to longstanding conjectures about high-dimensional percolation. For instance, precise information about the distribution of vertices within clusters and chemical distances between far away vertices would allow one to obtain the scaling limit of simple random walk on large critical percolation clusters [6]. We believe that many of the results and techniques that we obtain here could be useful for studying this and other open problems of the model.…”

Section: Introductionmentioning

confidence: 79%

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

Chatterjee¹,

Hanson²,

Sosoe³

2021

Preprint

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In high dimensional percolation at parameter p < p c , the one-arm probability π p (n) is known to decay exponentially on scale (p c − p) −1/2 . We show the same statement for the ratio π p (n)/π pc (n), establishing a form of a hypothesis of scaling theory.As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability p c . These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter n box on scale n d−6 ; this result complements a lower bound of Aizenman [1].

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Scaling Limit for the Ant in High‐Dimensional Labyrinths

Cited by 9 publications

References 44 publications

Random walk on barely supercritical branching random walk

Random walk on barely supercritical branching random walk

Efficient Methods for the Estimation of Homogenized Coefficients

Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

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