Infinite Canonical Super-Brownian Motion and Scaling Limits

Hofstad, Remco van der

doi:10.1007/s00220-006-0044-y

Cited by 11 publications

(20 citation statements)

References 60 publications

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“…The technique of lace expansion was developed to study critical random environments in high dimensions. It is expected that several models such as critical branching random walks, oriented percolation, percolation, and lattice trees have, in some sense, similar universal large-scale behavior as explained in section 6 of [52]. It is thus natural to expect that the simple random walk on all of those random environments should have similar limiting behaviors.…”

Section: Introductionmentioning

confidence: 94%

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: Introductionmentioning

confidence: 94%

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous

Cabezas

Fribergh

2019

Comm Pure Appl Math

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“…In [121], the natural conjecture was formulated that the scaling limit of the incipient infinite cluster for , and it is pointed out in [109] that this proves the conjecture at the level of convergence of finite-dimensional distributions.…”

Section: The Canonical Measure Of Sbmmentioning

confidence: 96%

“…A different but related and more general proof of (17.12) is discussed in [109]. The right hand side of (17.12) is equal toM (17.3).…”

Section: (1517)mentioning

confidence: 99%

The Lace Expansion and its Applications

Saint-Flour

Slade

Picard³

2006

Lecture Notes in Mathematics

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ii The Lace Expansion and its ApplicationsThe Lace Expansion and its Applications iii Preface Several superficially simple mathematical models, such as the self-avoiding walk and percolation, are paradigms for the study of critical phenomena in statistical mechanics. Although these models have been studied by mathematicians for about half a century, exciting new developments continue to occur and the subject is flourishing. Much progress has been made, but it remains a major challenge for mathematical physics and probability theory to obtain a complete and mathematically rigorous understanding of the scaling theory of these models at criticality.These lecture notes concern the lace expansion, which is a powerful tool for the analysis of the critical scaling of several models above their upper critical dimensions, namely:• the self-avoiding walk on Z d for d > 4, • lattice trees and lattice animals onResults include proofs of existence of critical exponents, with mean-field values, and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion. There are two distinct goals for these notes. The first goal is to provide a written accompaniment to my lectures at the Saint-Flour summer school in 2004, and at the Pacific Institute for the Mathematical Sciences -University of British Columbia summer school in 2005. The notes contain an introduction to the lace expansion and several of its applications, with sufficient background and depth to prepare a newcomer to do research using the lace expansion. Basic graduate level probability theory will be used, but no previous knowledge of the lace expansion or super-Brownian motion is assumed. The second goal is to provide a survey of the field, so that an interested reader can follow up by consulting the original literature. In pursuit of the second goal, these notes include more material than can be covered during a summer school course.Following a brief initial section concerning random walk, the notes can be divided into four parts, whose contents are summarized as follows.Part I, which concerns the self-avoiding walk, consists of Sections 2-6. A complete and self-contained proof is given of the convergence of the lace expansion for the nearest-neighbour model in dimensions d 4, and for the spread-out model of self-avoiding walks which take steps of length at most L, with L 1, in dimensions d > 4. The convergence proof presented here seems simpler than all previous lace expansion convergence proofs. As a consequence of convergence, it is shown that the critical exponent γ for the generating function of the number of n-step self-avoiding walks exists and is equal to 1. A survey is then given of the many extensions of this result that have been obtained using the lace expansion. Part II, which concerns lattice trees and lattice animals, consists of Sections 7-8. It is shown how a minor modification of the expansion for the self-avoiding walk can be applied to give expansions for lattice trees and lattice animals, and an indicatio...

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“…We follow the construction in [4] and [19]. Let W = ∞ n=0 {0} × N n be the set of finite words starting with a 0.…”

Section: Branching Random Walkmentioning

confidence: 99%

“…Recall that the survival probability of BRW satisfies nP(N n > 0) → 2/V (Kolmogorov [35]), and the r-point functions scale to those of SBM when the branching law has all moments; see, for example, [19] …”

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confidence: 99%

A criterion for convergence to super-Brownian motion on path space

Hofstad¹,

Holmes²,

Perkins³

2017

Ann. Probab.

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We give a sufficient condition for tightness for convergence of rescaled critical spatial structures to the canonical measure of super-Brownian motion. This condition is formulated in terms of the r-point functions for r = 2, . . . , 5. The r-point functions describe the expected number of particles at given times and spatial locations, and have been investigated in the literature for many high-dimensional statistical physics models, such as oriented percolation and the contact process above 4 dimensions and lattice trees above 8 dimensions. In these settings, convergence of the finite-dimensional distributions is known through an analysis of the r-point functions, but the lack of tightness has been an obstruction to proving convergence on path space.We apply our tightness condition first to critical branching random walk to illustrate the method as tightness here is well known. We then use it to prove tightness for sufficiently spread-out lattice trees above 8 dimensions, thus proving that the measure-valued process describing the distribution of mass as a function of time converges in distribution to the canonical measure of super-Brownian motion. We conjecture that the criterion will also apply to other statistical physics models.

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Infinite Canonical Super-Brownian Motion and Scaling Limits

Cited by 11 publications

References 60 publications

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Scaling Limit for the Ant in High‐Dimensional Labyrinths

The Lace Expansion and its Applications

A criterion for convergence to super-Brownian motion on path space

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