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We study a particular model of a random medium, called the orthant model, in general dimensions $dge 2$. Each site $xin Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,dots, d$ with probability $p$ and otherwise to its negative neighbours $x-e_i$, $i=1,dots, d$ (with probability $1-p$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when $p$ is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary, which is a key requirement of the subadditive ergodic theorem.
We prove a shape theorem for the set of infected individuals in a spatial epidemic model with 3 states (susceptible-infected-recovered) on ${mathbb Z}^d,dge 3$, when there is no extinction of the infection. For this, we derive percolation estimates (
We study the frog model on Cayley graphs of groups with polynomial growth rate $D geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one
In this paper we study the moderate deviations for the magnetization of critical Curie-Weiss model. Chen, Fang and Shao considered a similar problem for non-critical model by using Stein method. By direct and simple arguments based on Laplace method,
We consider a Moran model with two allelic types, mutation and selection. In this work, we study the behaviour of the proportion of fit individuals when the size of the population tends to infinity, without any rescaling of parameters or time. We fir
Any (measurable) function $K$ from $mathbb{R}^n$ to $mathbb{R}$ defines an operator $mathbf{K}$ acting on random variables $X$ by $mathbf{K}(X)=K(X_1, ldots, X_n)$, where the $X_j$ are independent copies of $X$. The main result of this paper concerns