• Corpus ID: 212628747

Absence of backward infinite paths for first-passage percolation in arbitrary dimension

@article{Brito2020AbsenceOB,
  title={Absence of backward infinite paths for first-passage percolation in arbitrary dimension},
  author={Gerandy Brito and Michael Damron and Jack Hanson},
  journal={arXiv: Probability},
  year={2020}
}
In first-passage percolation (FPP), one places nonnegative random variables (weights) $(t_e)$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on $\mathbb{Z}^d$ for $d \geq 2$ and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of… 
View PDF on arXiv
View With Semantic Reader
4 Citations
Background Citations
4

Figures from this paper

4 Citations

Citation Type

Citation Type

Empirical distributions, geodesic lengths, and a variational formula in first-passage percolation.
This article resolves, in a dense set of cases, several open problems concerning geodesics in i.i.d. first-passage percolation on $\mathbb{Z}^d$. Our primary interest is in the empirical measures of…
Non-existence of non-trivial bi-infinite geodesics in Geometric Last Passage Percolation
— We show non-existence of non-trivial bi-infinite geodesics in the solvable last-passage percolation model with i.i.d. geometric weights. This gives the first example of a model with discrete…
Random nearest neighbor graphs: the translation invariant case
If $(\omega(e))$ is a family of random variables (weights) assigned to the edges of $\mathbb{Z}^d$, the nearest neighbor graph is the directed graph induced by all edges $\langle x,y \rangle$ such…
Existence and Coexistence in First-Passage Percolation
We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in…

References

SHOWING 1-10 OF 34 REFERENCES
Busemann Functions and Infinite Geodesics in Two-Dimensional First-Passage Percolation
We study first-passage percolation on $${\mathbb{Z}^2}$$Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of…
Bigeodesics in First-Passage Percolation
In first-passage percolation, we place i.i.d. continuous weights at the edges of $${\mathbb{Z}^2}$$Z2 and consider the weighted graph metric. A distance-minimizing path between points x and y is…
Limiting geodesics for first-passage percolation on subsets of $\mathbb{Z}^{2}$
It is an open problem to show that in two-dimensional first-passage percolation, the sequence of finite geodesics from any point to $(n,0)$ has a limit in $n$. In this paper, we consider this…
Invariant percolation on trees and the mass-transport method
In bond percolation on an in nite locally nite graph G = (V;E), each edge is randomly assigned value 0 (absent) or 1 (present) according to some probability measure on f0; 1g. One then studies…
Geodesic Rays and Exponents in Ergodic Planar First Passage Percolation
We study first passage percolation on the plane for a family of invariant, ergodic measures on $\mathbb{Z}^2$. We prove that for all of these models the asymptotic shape is the $\ell$-$1$ ball and…
Nonexistence of Bigeodesics in Integrable Models of Last Passage Percolation
Bi-infinite geodesics are fundamental objects of interest in planar first passage percolation. A longstanding conjecture states that under mild conditions there are almost surely no bigeodesics,…
Random coalescing geodesics in first-passage percolation
We continue the study of infinite geodesics in planar first-passage percolation, pioneered by Newman in the mid 1990s. Building on more recent work of Hoffman, and Damron and Hanson, we develop an…
ABSENCE OF GEODESICS IN FIRST-PASSAGE PERCOLATION ON A HALF-PLANE
An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are…
Geodesics in first passage percolation
We consider a wide class of ergodic first passage percolation processes on I? and prove that there exist at least four one-sided geodesies a.s. We also show that coexistence is possible with positive…
Stationary coalescing walks on the lattice
We consider translation invariant measures on families of nearest-neighbor semi-infinite walks on the integer lattice. We assume that once walks meet, they coalesce. In 2d, we classify the collective…
...
1
2
3
4
...