Scaling limit for the ant in a simple high-dimensional labyrinth

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

doi:10.1007/s00440-018-0876-3

Cited by 11 publications

(13 citation statements)

References 37 publications

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“…Our main result, Theorem 1.1, can be seen as a scaling limit counterpart of the Alexander-Orbach conjecture (which identifies the scaling exponent). In particular, from our result, the exponent predicted by the Alexander-Orbach conjecture can be deduced, which, to the best of our knowledge, is a new result in the context of a simple random walk on critical branching random walks (see the companion article [13]).…”

Section: Discussing the Universality Of The Brownian Motion On The Issupporting

confidence: 58%

“…Even though we cannot prove, at this point, that they hold for bond percolation, we conjecture that they do. We illustrate the abstract theorem proved here in the companion paper [13]. There we prove that our four sufficient conditions hold in an interesting but simpler case, i.e., for the random walk on the trace of a large critical branching random walk in dimensions larger than 14. We thus obtain that the scaling limit is indeed the B ISE in this case.…”

Section: Introductionmentioning

confidence: 74%

“…The companion paper [13] provides an example of application of this abstract convergence theorem. There we show that the properly normalized random walk on the range of critical branching random walks converges to the Brownian motion on the ISE.…”

Section: Introductionmentioning

confidence: 99%

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Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous

Cabezas

Fribergh

2019

Comm Pure Appl Math

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We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super‐Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread‐out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14. © 2019 Wiley Periodicals, Inc.

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Section: Discussing the Universality Of The Brownian Motion On The Issupporting

confidence: 58%

Section: Introductionmentioning

confidence: 74%

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Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous

Cabezas

Fribergh

2019

Comm Pure Appl Math

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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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“…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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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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Scaling limit for the ant in a simple high-dimensional labyrinth

Cited by 11 publications

References 37 publications

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Efficient Methods for the Estimation of Homogenized Coefficients

Random walk on barely supercritical branching random walk

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