Related papers: Sublinear variance in first-passage percolation fo…

In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value…

We consider first-passage percolation with positive, stationary-ergodic weights on the square lattice $\mathbb{Z}^d$. Let $T(x)$ be the first-passage time from the origin to a point $x$ in $\mathbb{Z}^d$. The convergence of the scaled…

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…

Consider first passage percolation on $\mathbb{Z}^d$ with passage times given by i.i.d. random variables with common distribution $F$. Let $t_\pi(u,v)$ be the time from $u$ to $v$ for a path $\pi$ and $t(u,v)$ the minimal time among all…

Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on…

The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage…

We investigate a novel first-passage percolation model, referred to as the Brochette first-passage percolation model, where the passage times associated with edges lying on the same line are equal. First, we establish a point-to-point…

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [6]. We describe our results…

We determine the asymptotic speed of the first-passage percolation process on some ladder-like graphs (or width-2 stretches) when the times associated with different edges are independent and exponentially distributed but not necessarily…

We introduce and study derivatives in first-passage percolation with edge weights given by i.i.d. random variables supported on ${a,b}$. We show that the variance of the passage time can be expressed in terms of these derivatives. We…

We provide a new exponential concentration inequality for First Passage Percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm which gave a variance bound for…

We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the origin to the point $\big(n,n^{\lfloor a…

We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. The passage time of edge $e$ is $0$ with probability $p$ and $1$ with probability $1-p$, independently of each other. Let $p_c$ be…

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits…

We consider the standard first passage percolation model on $\mathbb Z^d$ with bounded and bounded away from zero weights. We show that the rescaled passage time $\widetilde{\mathbf T}_{n,X}$ restricted to a compact set $X$ satisfies a…

We study the first passage times of discrete-time branching random walks in ${\mathbb R}^d$ where $d\geq 1$. Here, the genealogy of the particles follows a supercritical Galton-Watson process. We provide asymptotics of the first passage…

First-passage percolation is the study of the metric space $(\mathbb{Z}^d,T)$, where $T$ is a random metric defined as the weighted graph metric using random edge-weights $(t_e)_{e\in \mathcal{E}^d}$ assigned to the nearest-neighbor edges…

This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential…

We consider a first-passage percolation model on a Delaunay triangulation of the plane. In this model each edge is independently equipped with a nonnegative random variable, with distribution function F, which is interpreted as the time it…

Let $0<a<b<\infty$ be fixed scalars. Assign independently to each edge in the lattice $\mathbb{Z}^2$ the value $a$ with probability $p$ or the value $b$ with probability $1-p$. For all $u,v\in\mathbb{Z}^2$, let $T(u,v)$ denote the first…