First passage percolation: The stationary case

Boivin, Daniel

doi:10.1007/bf01198171

Cited by 50 publications

(71 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The extension of the Cox and Durrett [10] theorem to Z d was shown by Kesten [21]. Boivin [7] proved the result for stationaryergodic media. Despite these strong existence results on the time-constant and limit-shape, surprisingly little else is known in sufficient generality [39].…”

Section: Background On the Time-constantmentioning

confidence: 89%

Variational Formula for the Time Constant of First‐Passage Percolation

Krishnan

2016

Comm Pure Appl Math

View full text Add to dashboard Cite

We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ℤd. Let T(x) be the first‐passage time from the origin to a point x in ℤd. The convergence of the scaled first‐passage time T([nx])/n to the time constant as n → ∞ can be viewed as a problem of homogenization for a discrete Hamilton‐Jacobi‐Bellman (HJB) equation. We derive an exact variational formula for the time constant and construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption). We explicitly identify when the iteration produces correctors.© 2016 Wiley Periodicals, Inc.

show abstract

Section: Background On the Time-constantmentioning

confidence: 89%

Variational Formula for the Time Constant of First‐Passage Percolation

Krishnan

2016

Comm Pure Appl Math

View full text Add to dashboard Cite

show abstract

“…Thanks to Lemma 3.5, (3) and (4) respectively converge to µ(x+y)P p (0 ↔ ∞) 3 and µ(x)P p (0 ↔ ∞) 3 . Moreover,…”

Section: Lemma 34 Let X and Y Be Two Linearly Independent Vectors Inmentioning

confidence: 99%

“…We use Lemma 3.5 again to conclude that (5) converges to µ(y)P p (0 ↔ ∞) 3 . As P p (0 ↔ ∞) > 0, this leads to the desired inequality.…”

Section: Lemma 34 Let X and Y Be Two Linearly Independent Vectors Inmentioning

confidence: 99%

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Garet

Marchand

2004

ESAIM: PS

View full text Add to dashboard Cite

Abstract. The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on Z d to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given.Mathematics Subject Classification. 60G15, 60K35, 82B43. When the passage times are independent identically distributed variables, Cox and Durrett [7] showed that, under some moment conditions, the renormalized set of wet vertices at time t almost surely converges to a deterministic asymptotic shape. Derriennic (cited by Kesten [13]), and next Boivin [3], progressively extended the result to the stationary ergodic case. Häggström and Meester [11] also proved that every symmetric compact set with nonempty interior can be obtained as the asymptotic shape of a stationary first-passage percolation model.In this paper, we want to study the analogous problem of spread of a fluid in a more complex medium. On one hand, an edge can either be open or closed according to the local properties of the medium -e.g. the absence or the presence of non-porous particles. In other words, the Z d lattice is replaced by a random environment given by the infinite cluster of a super-critical Bernoulli percolation model. On the other hand, as in the classical model, a random passage time is attached to each open edge. This random time corresponds to the local porosity of the medium -e.g. the density of the porous phase. Thus, our model can be seen as a combination between classical Bernoulli percolation and stationary first-passage percolation.

show abstract

“…The limit shape referred to above is the set B ⊂ R 2 given by B = {z : g(z) ≤ 1}, and g is the asymptotic norm for the FPP model (see [ 2 . B is also characterized by the "shape theorem," (see [3,Theorem 2.6] and [6]) which says that given ǫ > 0, one has P ((1 − ǫ)B ⊂ B(t)/t ⊂ (1 + ǫ)B for all large t) = 1, where B(t)/t is the set {z/t : z ∈ B(t)}. There is no simple condition known to guarantee that the limit shape under condition A2' is bounded.…”

Section: Assumptionsmentioning

confidence: 99%

Bigeodesics in First-Passage Percolation

Damron

Hanson

2016

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

First passage percolation: The stationary case

Abstract: If the passage time of the edges of the 2g a lattice are stationary, ergodic and have finite moment of order p > d, then a.s. the set of vertices that can be reached within time t, has an asymptotic shape as t--* oe.

Cited by 50 publications

References 9 publications

Variational Formula for the Time Constant of First‐Passage Percolation

Variational Formula for the Time Constant of First‐Passage Percolation

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

Bigeodesics in First-Passage Percolation

Contact Info

Product

Resources

About