Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

Kumagai, Takashi; Misumi, Jun

doi:10.1007/s10959-008-0183-5

Cited by 65 publications

(125 citation statements)

References 19 publications

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“…This will be done via an appeal to the following theorem of Barlow, Járai, Kumagai, and Slade [12], which gives a sufficient condition for Alexander-Orbach behaviour. See [12] for quantitative versions of the theorem, and [48] for generalizations. We recall that the effective conductance between two disjoint finite sets A, B in a finite network G is defined to be The estimate (8.1) has already been established in Corollary 6.3 and Lemma 6.11.…”

Section: Spectral Dimension Anomalous Diffusionmentioning

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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Section: Spectral Dimension Anomalous Diffusionmentioning

confidence: 99%

Universality of high-dimensional spanning forests and sandpiles

Hutchcroft

2019

Probab. Theory Relat. Fields

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“…In this light, our abstract convergence theorem, Theorem 1.1, can be seen as a more delicate version of [47], which gave sufficient conditions to prove the Alexander-Orbach conjecture that were subsequently applied to prove that conjecture in the case of critical percolation (see [42,45] for a generalization).…”

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

confidence: 99%

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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“…Since in our case |E(G)| ≈ r 2 we get that the commute time is ≈ r 3 . Now, in general the commute time only bounds the hitting time Hit(0, ∂B IIC (0, r)) from above, but in strongly recurrent graphs this turns out to be sharp [39]. Thus, in time r 3 the random walk has walked only in B IIC (0, r) and it can be shown that the end point is approximately uniformly distributed (the walk has mixed in B IIC (0, r) in that time).…”

Section: Introductionmentioning

confidence: 99%

The Alexander-Orbach conjecture holds in high dimensions

2009

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Abstract. We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d > 6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension, that is, pt(x, x) = t −2/3+o(1) . This establishes a conjecture of Alexander and Orbach [4]. En route we calculate the one-arm exponent with respect to the intrinsic distance.

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Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

Cited by 65 publications

References 19 publications

Universality of high-dimensional spanning forests and sandpiles

Universality of high-dimensional spanning forests and sandpiles

Scaling Limit for the Ant in High‐Dimensional Labyrinths

The Alexander-Orbach conjecture holds in high dimensions

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