We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost…

We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. The present counterexample only works for first-passage percolation on…

This work was motivated by the recent work by H. Dette, J. Pitman and W.J. Studden on a new duality relation for random walks. In this note we consider the diffusion process limit of their result, and use the alternative…

The three-color Greenberg--Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice $Z^2$ is initially assigned one of the states 0, 1 or 2. At each tick of a discrete--time clock, the…

Consider a countable collection of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium fluctuating both in space and time. Here we study the…

We study the $d$-dimensional branching random walk for $d>2$. This process has extremal equilibria for every intensity. We are interested in the large space scale and large space-time scale behavior of the equilibrium state. We show that the fluctuations of…

The law for the time $\tau_{a}$ spent by a standard Brownian excursion above a given level $a > 0$ is found using Ito excursion theory. This is achieved by conditioning the excursion to have exactly one mark of an independent…