Multiple geodesics with the same direction

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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“…From this, one can show absence of bigeodesics in deterministic directions. Further work was done by Coupier in [9].…”

Section: Geodesics In Related Modelsmentioning

confidence: 99%

Bigeodesics in First-Passage Percolation

Damron

Hanson

2016

Commun. Math. Phys.

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“…It is well known that any stationary measure with respect to (5) that is also translation invariant is a convex combination of {ν α : 0 ≤ α ≤ 1} (see [15]). Another way to describe the TASEP is through the so-called Harris construction.…”

Section: Define the Measures {νmentioning

confidence: 99%

“…Illustration of the sorting process dynamics. The particle (5, 7) interacts with the only particle that is not ordered with respect to, (4,8), to create two new particles -(4, 7) and (5,8).…”

Section: Stationarity Of the Tazrp Speed Processmentioning

confidence: 99%

The TASEP speed process

Amir¹,

Angel²,

Valkó³

2011

Ann. Probab.

102

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In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of Z, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.Comment: Published in at http://dx.doi.org/10.1214/10-AOP561 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…In recent years there has been a great deal of interest in studying the coalescence of polymers (maximal paths which we shall also refer to as geodesics) in last passage percolation models as well [19,15,36]. Much can be established in certain exactly solvable settings including the existence and uniqueness of semi-infinite geodesics starting at a given point along a given direction, and coalescence of geodesics along deterministic directions.…”

Section: Introductionmentioning

confidence: 99%

Coalescence of geodesics in exactly solvable models of last passage percolation

Basu

Hoffman

Sly

2019

Journal of Mathematical Physics

121

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Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z 2 with i.i.d. exponential weights on the vertices. Fix two points v 1 = (0, 0) and v 2

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Multiple geodesics with the same direction

Cited by 34 publications

References 10 publications

Bigeodesics in First-Passage Percolation

Bigeodesics in First-Passage Percolation

The TASEP speed process

Coalescence of geodesics in exactly solvable models of last passage percolation

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