Bigeodesics in First-Passage Percolation

  title={Bigeodesics in First-Passage Percolation},
  author={Michael Damron and Jack Hanson},
  journal={Communications in Mathematical Physics},
  • M. Damron, J. Hanson
  • Published 2 December 2015
  • Mathematics
  • Communications in Mathematical Physics
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 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 1990s, Licea–Newman showed that, under a curvature assumption on the “asymptotic shape,” all infinite geodesics have an asymptotic… 
Geodesics, bigeodesics, and coalescence in first passage percolation in general dimension
We consider geodesics for first passage percolation (FPP) on $\mathbb{Z}^d$ with iid passage times. As has been common in the literature, we assume that the FPP system satisfies certain basic
Absence of backward infinite paths for first-passage percolation in arbitrary dimension
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 $\mathbb{Z}^d$
Optimal exponent for coalescence of finite geodesics in exponential last passage percolation
In this note, we study the model of directed last passage percolation on $\mathbb{Z}^2$, with i.i.d. exponential weight. We consider the maximum paths from vertices $\left(0,\left\lfloor k^{2/3}
Geodesics Toward Corners in First Passage Percolation
For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and
Coalescence of geodesics in exactly solvable models of last passage percolation
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
First Passage Percolation on Hyperbolic groups
We study first passage percolation (FPP) on a Gromov-hyperbolic group $G$ with boundary $\partial G$ equipped with the Patterson-Sullivan measure $\nu$. We associate an i.i.d.\ collection of random
Fluctuations of transverse increments in two-dimensional first passage percolation
We consider finite geodesics for first passage percolation (FPP) on $\mathbb{Z}^2$ with i.i.d.\ continuous passage times having exponential moments. As has been common in the literature, we assume
Existence and coalescence of directed infinite geodesics in the percolation cone
In [HN01], Howard and Newman gave an essentially complete picture about existence and uniqueness of asymptotically directed infinite geodesics on Poisson point process on $\mathbb{R}^d$. For $d=2$,
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,
Geometry of geodesics through Busemann measures in directed last-passage percolation
We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of


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
Geodesics and Recurrence of Random Walks in Disordered Systems
In a first-passage percolation model on the square lattice $Z^2$, if the passage times are independent then the number of geodesics is either $0$ or $+\infty$. If the passage times are stationary,
Geodesics and spanning trees for Euclidean first-passage percolation
The metric D α (q, q') on the set Q of particle locations of a homogeneous Poisson process on R d , defined as the infimum of (Σ i |q i -q i+1 | α ) 1/α over sequences in Q starting with q and ending
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 and the competition interface for the corner growth model
We study the directed last-passage percolation model on the planar integer lattice with nearest-neighbor steps and general i.i.d. weights on the vertices, outside the class of exactly solvable
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
A shape theorem and semi-infinite geodesics for the Hammersley model with random weights
In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also
50 years of first passage percolation
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
A Surface View of First-Passage Percolation
Let \(\tilde B\)(t) be the set of sites reached from the origin by time t in standard first-passage percolation on Z d , and let B0 (roughly lim \(\tilde B\) (t)/t) be its deterministic asymptotic
Geodesics in two-dimensional first-passage percolation
We consider standard first-passage percolation on Z 2 . Geodesics are nearest-neighbor paths in Z 2 , each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite