Collisions of random walks in dynamic random environments

Halberstam, Noah; Hutchcroft, Tom

doi:10.1214/21-ejp738

Cited by 5 publications

(4 citation statements)

References 35 publications

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“…The goal of this subsection is to prove the following general proposition concerning intersections of random walks on general unimodular random rooted graphs. The proof of this proposition is of a similar flavour to those of [25,29], which involve collisions (where the two walks are at the same location at the same time) rather than intersections (where the two walks are at the same location but not necessarily at the same time).…”

Section: A Criterion For the Infinite Intersection Propertymentioning

confidence: 79%

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Uniqueness of the infinite tree in low-dimensional random forests

Halberstam¹,

Hutchcroft²

2023

Preprint

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The arboreal gas is the random (unrooted) spanning forest of a graph in which each forest is sampled with probability proportional to β #edges for some β ≥ 0, which arises as the q → 0 limit of the Fortuin-Kastelyn random cluster model with p = βq. We study the infinite-volume limits of the arboreal gas on the hypercubic lattice Z d , and prove that when d ≤ 4, any translation-invariant infinite volume Gibbs measure contains at most one infinite tree almost surely. Together with the existence theorem of Bauerschmidt, Crawford and Helmuth (2021), this establishes that for d = 3, 4 there exists a value of β above which subsequential weak limits of the β-arboreal gas on tori have exactly one infinite tree almost surely. We also show that the infinite trees of any translation-invariant Gibbs measure on Z d are one-ended almost surely in every dimension. The proof has two main ingredients: First, we prove a resampling property for translation-invariant arboreal gas Gibbs measures in every dimension, stating that the restriction of the arboreal gas to the trace of the union of its infinite trees is distributed as the uniform spanning forest on this same trace. Second, we prove that the uniform spanning forest of any translation-invariant random connected subgraph of Z d is connected almost surely when d ≤ 4. This proof also provides strong heuristic evidence for the conjecture that the supercritical arboreal gas contains infinitely many infinite trees in dimensions d ≥ 5. Along the way, we give the first systematic and axiomatic treatment of Gibbs measures for models of this form including the random cluster model and the uniform spanning tree.

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Section: A Criterion For the Infinite Intersection Propertymentioning

confidence: 79%

“…We state a simple special case of the relevant theorem now, with a significant generalization given in Theorem 4.2. The proof of this theorem draws mostly on random walk techniques, and is inspired in particular by previous work on collisions of random walks in unimodular random graphs [25,29].…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of the infinite tree in low-dimensional random forests

Halberstam¹,

Hutchcroft²

2023

Preprint

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“…The code used to generate all the simulations and the associated data presented in the paper is available at and in the electronic supplementary material [77].…”

Section: Data Accessibilitymentioning

confidence: 99%

What are the limits of universality?

Halberstam

Hutchcroft

2022

Proc. R. Soc. A.

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It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents on Z 4 and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent τ . This gives strong evidence against a conjecture of Balankin et al. (2018 Phys. Lett. A 382 , 12–19 ( doi:10.1016/j.physleta.2017.10.035 )).

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“…First mentioned in Pólya's note [18], the collision of random walks has since then been a classic topic in probability theory. Recently, this topic has gained more attention from researchers working on the random walks on graphs [4,12] and random environments [11,9,3]. When consider only two random walks, collision problems are strongly related to Brownian local time [15,19,23].…”

Section: Introductionmentioning

confidence: 99%

Scaling limit of the collision measures of multiple random walks

Nguyen¹

2022

Preprint

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For an integer k ≥ 2, let S (1) , S (2) , . . . , S (k) be k independent simple symmetric random walks on Z. A pair (n, z) is called a collision event if there are at least two distinct random walks, namely, S (i) , S (j) satisfyingWe show that under the same scaling as in Donsker's theorem, the sequence of random measures representing these collision events converges to a non-trivial random measure on [0, 1] × R. Moreover, the limit random measure can be characterized using Wiener chaos. The proof is inspired by methods from statistical mechanics, especially, by a partition function that has been developed for the study of directed polymers in random environments.

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Collisions of random walks in dynamic random environments

Cited by 5 publications

References 35 publications

Uniqueness of the infinite tree in low-dimensional random forests

Uniqueness of the infinite tree in low-dimensional random forests

What are the limits of universality?

Scaling limit of the collision measures of multiple random walks

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