The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on Z to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As...

We consider lattice spin systems with short range but random and unbounded interactions. Statics : We give an elementary proof of uniqueness of Gibbs measures at high temperature or strong magnetic elds, and of the exponential decay of the corresponding quenched correlation functions. The analysis is based on the study of disagreement percolation (as initiated in van den Berg{Maes (1994)). Dyna...

We prove that there exist natural generalizations of the classical bootstrap percolation model on Z that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter [28] (in the case d = r = 2) and Schonmann [25] (for all d > r > 2) proved that r-neighbour bootstrap percolation models have trivial critical probabili...

Boolean networks, widely used to model gene regulation, exhibit a phase transition between regimes in which small perturbations either die out or grow exponentially. We show and numerically verify that this phase transition in the dynamics can be mapped onto a static percolation problem which predicts the long-time average Hamming distance between perturbed and unperturbed orbits.

Optimal flow paths obtained from percolation theory provide a powerful tool that can be used to characterize properties associated with flow such as soil hydraulic conductivity, as well as other properties influenced by flow connectivity and topology. A recently proposed scaling theory for vegetation growth appeals to the tortuosity of optimal paths from percolation theory to define the spatio-...

We present a parallel algorithm for finding the shortest path whose total weight is smaller than a pre-determined value. The passage times over the edges are assumed to be positive integers. In each step the processing elements are not analyzing the entire graph. Instead they are focusing on a subset of vertices called active vertices. The set of active vertices at time t is related to the boun...

We discuss the spatiotemporal intermittency (STI) seen in coupled map lattices (CML-s). We identify the types of intermittency seen in such systems in the context of several specific CML-s. The Chaté-Manneville CML is introduced and the on-going debate on the connection of the spatiotemporal intermittency seen in this model with the problem of directed percolation is summarised. We also discuss...

We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probability p and 1 p, respectively. We prove that as the zero-time bonds approach the percolation threshold p~, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeys ~(p) ~ ...

In 2-neighborhood bootstrap percolation on a graphG, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least two already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to e...

Self-avoiding walks, lattice trees and lattice animals, and percolation are among the simplest models exhibiting the general features of critical phenomena. A basic problem is to prove the existence of critical exponents governing their behavior near the critical point. This problem gains importance from interrelations between these models and models of ferromagnetism such as the Ising model, a...