Capacity of the range of random walk on $\mathbb {Z}^d$

Asselah, Amine; Schapira, Bruno; Sousi, Perla

doi:10.1090/tran/7247

Cited by 36 publications

(64 citation statements)

References 12 publications

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“…=⇒ Cap(β[0, 1]), (1.5) with σ some renormalising constant and β[0, 1] the trace of a three-dimensional Brownian motion between time 0 and 1. In [4], we also proved a standard central limit theorem in dimension larger than or equal to 6 (with a standard √ n normalising factor, and a Gaussian limit), while the law of large numbers had already been obtained in dimension 5 and larger by Jain and Orey [18], almost fifty years ago. A striking correspondence emerges: all these results for the capacity of the range are analogous to results for the volume of the range, see [13,19,28], but only after dropping space dimension by two units to go from capacity to volume of the range.…”

Section: Introductionmentioning

confidence: 61%

Capacity of the range of random walk on $\mathbb{Z}^{4}$

Asselah¹,

Schapira²,

Sousi³

2019

Ann. Probab.

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We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 [28] for the volume of the range in dimension two.One easily passes from one representation in (1.1) to the other using the last passage decomposition formula, see (2.8) below. Denote by {S(n), n ∈ N} a simple random walk in Z 4 . For two integers m, n, the range R[m, n] (or simply R n when m = 0) in the time period [m, n] is defined as R[m, n] = {S(m), . . . , S(n)}.Our first result is a strong law of large numbers for Cap (R n ).Recently, van den Berg, Bolthausen and den Hollander [38] advocated a new geometric characteristic, the torsional rigidity of the complement of a Wiener sausage, as a way to probe the shape of the sausage. In order to obtain leading asymptotics for the torsional rigidity, one needs a law of large numbers for the capacity of a Wiener sausage, which is not proved yet in dimension four; see however our companion paper [5] for a partial result in this direction. Our Theorem 1.1 establishes these asymptotics for the discrete model, and thus prepares the study of torsional rigidity for random walk.

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Section: Introductionmentioning

confidence: 61%

Capacity of the range of random walk on $\mathbb{Z}^{4}$

Asselah¹,

Schapira²,

Sousi³

2019

Ann. Probab.

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“…Lemmas 5.5 and 5.6 give the correct order of magnitude for the capacity of the random walk on Z d for all d ≥ 3, which is order √ n when d = 3, order n/ log n when d = 4, and order n when d ≥ 5. See [5] and references therein for more detailed results. Lemma 5.6 has the following immediate corollary.…”

Section: Note That For Any Two Sets Of Verticesmentioning

confidence: 99%

Universality of high-dimensional spanning forests and sandpiles

Hutchcroft

2019

Probab. Theory Relat. Fields

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We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of Z d , d ≥ 5. In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph.In the case of Z d , d ≥ 5, some of our results regarding critical exponents recover earlier results of Bhupatiraju, Hanson, and Járai (Electron. J. Probab. Volume 22 (2017), no. 85 ). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work. * Statslab, DPMMS, University of Cambridge

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“…The analogous result for the volume of the sausage in d = 2 (or even the size of the range of a random walk) is not known. On the other hand, the correct order for the variance should be t 2 /(log t) 4 , as was proved in the discrete setting [3]. Thus our bound in (1.9) is off only by a (log log t) 9 term.…”

Section: Introductionmentioning

confidence: 62%

“…which should not come as a surprise, since this formula holds for deterministic sets (1.2) (but one still need to justify the interchange of limit and expectation). We next introduce the following stopping time 4) and note that the probability on the right-hand side of (3.3) is just the probability of τ being finite.…”

Section: Statement Of the Results And Sketch Of Proofmentioning

confidence: 99%

“…Variance Estimates. We choose L such that (log t) 4 ≤ 2 L ≤ 2(log t) 4 , so that L is of order log log t. Let now ε > 0 be gixed. By (2.3) and Chebychev's inequality, for t large enough, P |Cap(W 1 [0, t])| > ε t log t ≤ P (W 1 [0, t] ⊂ B(0, r)) + P |Υ(t, L, r)| > ε 2 t log t + P |S(t, L) − Ξ(t, L, r)| > ε 2 t log t ≤ e −c(log t) 2 + P |Υ(t, L, r)| > ε 2 t log t + 8(log t) 2 var(S(t, L)) + var(Ξ(t, L, r)) ε 2 t 2 .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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