Related papers: Geodesics in First Passage Percolation

We study first-passage percolation on Z2, where the edge weights are given by a translation-ergodic distribution, addressing questions related to existence and coalescence of infinite geodesics. Some of these were studied in the late 90's…

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…

We study first-passage percolation through related optimization problems over paths of restricted length. The path length variable is in duality with a shift of the weights. This puts into a convex duality framework old observations about…

The metric $D_\alpha (q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^\alpha)^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$…

For first passage percolation (FPP) on integer lattice with i.i.d. passage time distributions, in order to show existence of semi-infinite geodesics along a fixed direction, one requires unproven assumptions on the limiting shape. We…

In this paper, we study some properties of optimal paths in the first passage percolation on $\Z^d$ and show the followings: (1) the number of optimal paths has an exponential growth if the distribution has an atom; (2) the means of…

We establish the scaling limit of the geodesics to the root for the first passage percolation distance on random planar maps. We first describe the scaling limit of the number of faces along the geodesics. This result enables us to compare…

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 passage times to each edge of a Cayley…

There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$…

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first…

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit) for the…

This paper is a survey of various results and techniques in first passage percolation, a random process modeling a spreading fluid on an infinite graph. The latter half of the paper focuses on the connection between first passage…

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 ergodic theory for infinite geodesics via…

First passage percolation with recovery is a process aimed at modeling the spread of epidemics. On a graph $G$ place a red particle at a reference vertex $o$ and colorless particles (seeds) at all other vertices. The red particle starts…

In the classic model of first passage percolation, for pairs of vertices separated by a Euclidean distance $L$, geodesics exhibit deviations from their mean length $L$ that are of order $L^\chi$, while the transversal fluctuations, known as…

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, however the result has not been proved in any…

We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape…

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…

We study first-passage percolation on $\mathbb Z^d$, $d\ge 2$, with independent weights whose common distribution is compactly supported in $(0,\infty)$ with a uniformly-positive density. Given $\epsilon>0$ and $v\in\mathbb Z^d$, which…

We consider standard first-passage percolation on $\Z^d$. Let $e_1$ be the first coordinate vector. Let $a(n)$ be the expected passage time from the origin to $ne_1$. In this short paper, we note that $a(n)$ is increasing under some strong…