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 consider the stirring process in the interval $\Lambda_N:=[-N,N]$ of $\mathbb Z$ with  births and deaths taking place in the intervals $I_+:=(N-K,N]$, and respectively $I_-:=[-N,-N+K)$, $1 \le K <N$. We prove bounds on the truncated moments uniform in $N$ which…

We derive the following property of the "true self-repelling motion", a continuous real-valued self-interacting process $(X_t, t \ge 0)$ introduced by Balint Toth and Wendelin Werner. Conditionally on its occupation time measure at time one (which is the information about…

We consider exclusion process with degenerate rates  in a finite torus with size $n$. This model is a simplified model for some peculiar phenomena of the "glassy" dynamics. We prove that the spectral gap is bounded below by $C\rho^4/n^2$, where $\rho =…

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…

Let $S_{n,k}$ denote the random geometric graph obtained by placing points inside a square of area $n$ according to a Poisson point process of intensity $1$ and joining each such point to the $k=k(n)$ points of the process nearest to…