Geodesics and Recurrence of Random Walks in Disordered Systems

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2 + log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

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“…This theorem has been extended to some non-i.i.d. times by Boivin-Derrien [28]. In that paper, they also construct non-i.i.d.…”

Section: Rigorous Resultsmentioning

confidence: 99%

50 Years of First-Passage Percolation

Auffinger

Damron

Hanson

2017

University Lecture Series

227

427

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“…In fact, behavior for geodesics in some stationary models that is quite different from that predicted in the i.i.d. case has already been displayed; see [7] for an example with exactly one bigeodesic. So for such models, item 2 above would be false, and item 1 would not give any information.…”

Section: Previous Results On Geodesicsmentioning

confidence: 99%

Bigeodesics in First-Passage Percolation

Damron

Hanson

2016

Commun. Math. Phys.

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In first-passage percolation, we place i.i.d. continuous weights at the edges of Z 2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed that, under a curvature assumption on the "asymptotic shape," all infinite geodesics have an asymptotic direction, and there is a full measure set D ⊂ [0, 2π) such that for any θ ∈ D, there are no bigeodesics with one end directed in direction θ. In this paper, we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction. Furthermore, it resolves the BenjaminiKalai-Schramm "midpoint problem" [5, p. 1976] under the extra assumption that the limit shape boundary is differentiable.

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“…In other words, if the undirected version of G has at least three infinite components, then G must exhibit bi-infinite trajectories. It is perhaps worth noting that the dichotomy breaks down in other ways without assumption c): for instance, see [3] for a model which exhibits a bi-infinite trajectory and also exhibits coalescence. Perhaps the techniques used to prove Theorem 1.3 can be used to completely classify the allowed behavior of stationary coalescing walks which do not necessarily pass hyperplanes.…”

Section: Implications For Coalescing Walksmentioning

confidence: 99%

“…(See[17, Figs. 2,3] for illustrations of the structure of G 0 .) Although G 0 is not translation invariant, we can remedy this by letting U be an independent uniform vector on the set {0, e 1 , e 2 , e 1 + e 2 } and setting G to be the translation of G 0 by U .…”

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confidence: 99%

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Random nearest neighbor graphs: the translation invariant case

Bock¹,

Damron²,

Hanson³

2020

Preprint

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If (ω(e)) is a family of random variables (weights) assigned to the edges of Z d , the nearest neighbor graph is the directed graph induced by all edges x, y such that ω({x, y}) is minimal among all neighbors y of x. That is, each vertex points to its closest neighbor, if the weights are viewed as edge-lengths. Nanda-Newman introduced nearest neighbor graphs when the weights are i.i.d. and continuously distributed and proved that a.s., all components of the undirected version of the graph are finite. We study the case of translation invariant, distinct weights, and prove that nearest neighbor graphs do not contain doubly-infinite directed paths. In contrast to the i.i.d. case, we show that in this stationary case, the graphs can contain either one or two infinite components (but not more) in dimension two, and k infinite components for any k ∈ [1, ∞] in dimension ≥ 3. The latter constructions use a general procedure to exhibit a certain class of directed graphs as nearest neighbor graphs with distinct weights, and thereby characterize all translation invariant nearest neighbor graphs. We also discuss relations to geodesic graphs from first-passage percolation and implications for the coalescing walk model of Chaika-Krishnan.

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Geodesics and Recurrence of Random Walks in Disordered Systems

Cited by 5 publications

References 15 publications

50 Years of First-Passage Percolation

50 Years of First-Passage Percolation

Bigeodesics in First-Passage Percolation

Random nearest neighbor graphs: the translation invariant case

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