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 investigate the evolution of the random interfaces in a two dimensional Potts model at zero temperature under Glauber dynamics for some particular initial conditions. We prove that under space-time diffusive scaling the shape of the interfaces converges in probability…

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as…

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system…

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…