Related papers: Monotonicity in first-passage percolation
We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on $N$ vertices. The processes are allowed to spread with different rates, start from vertex subsets of different…
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 first-passage percolation and study their…
We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves…
We consider first passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen…
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…
The Ornstein-Uhlenbeck process of diffusion in the harmonic potential is re-examined in the context of the first-passage time problem. We investigate this problem to the extent that it has not yet been fully resolved and demonstrate exact…
We study the problem of coexistence in a two-type competition model governed by first-passage percolation on $\Zd$ or on the infinite cluster in Bernoulli percolation. Actually, we prove for a large class of ergodic stationary passage times…
Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage…
Recently, many results have been established drawing a parallel between Bernoulli percolation and models given by levels of smooth Gaussian fields with unbounded, strongly decaying correlation. In a previous work with D. Gayet , we started…
We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb…
Let $\{X(v), v \in \Bbb Z^d \times \Bbb Z_+\}$ be an i.i.d. family of random variables such that $P\{X(v)= e^b\}=1-P\{X(v)= 1\} = p$ for some $b>0$. We consider paths $\pi \subset \Bbb Z^d \times \Bbb Z_+$ starting at the origin and with…
Intuition suggests that passage times across a region increases with the number of barriers along the path. Can this fail depending on the nature of the barrier? To probe this fundamental question, we exactly solve for the first passage…
We study a natural growth process with competition, modeled by two first passage percolation processes, $FPP_1$ and $FPP_\lambda$, spreading on a graph. $FPP_1$ starts at the origin and spreads at rate $1$, whereas $FPP_\lambda$ starts from…
Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed…
We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $\mu(\mathbf{u})$ the time…
This paper focuses on the time constant for last passage percolation on complete graph. Let $G_n=([n],E_n)$ be the complete graph on vertex set $[n]=\{1,2,\ldots,n\}$, and i.i.d. sequence $\{X_e:e\in E_n\}$ be the passage times of edges.…
Tree models for rigidity percolation are introduced and solved. A probability vector describes the propagation of rigidity outward from a rigid border. All components of this ``vector order parameter'' are singular at the same rigidity…
We study the shape fluctuation in the first passage percolation on $\mathbb{Z}^d$. It is known that it diverges when the distribution obeys Bernoulli in [Yu Zhang. The divergence of fluctuations for shape in first passage percolation.…
We develop a model to compute the first-passage time of a random walker in a crowded environment. Hard-core particles with the same size and diffusion coefficient than the tracer diffuse, and the model allows to compute the first passage…
We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $\phi_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law…