The shape theorem for the frog model

Abstract

In this work we prove a shape theorem for a growing set of Simple Random Walks (SRWs) on Z d , known as frog model. The dynamics of this process is described as follows: There are active particles, which perform independent SRWs, and sleeping particles, which do not move. When a sleeping particle is hit by an active particle, it becomes active too. At time 0 all particles are sleeping, except for that placed at the origin. We prove that the set of the original positions of all the active particles, rescaled by the elapsed time, converges to some compact convex set. In some specific cases we are able to identify (at least partially) this set.

Local and Global Survival for Nonhomogeneous Random Walk Systems on Z

“…Theorem 2.7 yields the conclusion. Other, more powerful, conditions can be derived from Theorem 3.5 (2) and (3) by taking p n = 1 for all n ∈ N.…”

“…We note in particular that for conditions (2) and (3) we do not need the independence of {l n } n∈N .…”

“…(1) =⇒ (2). Suppose that (1) holds and there exists r 0 ∈ N such that, for all n ≥ r 0 we have α n > 0.…”

“…This result has been extended in [17] to the case of a random initial configuration (d ≥ 3) and in [9] for random walks on Z with right drift. Shape theorems on Z d can be found in [2,3]. Phase transitions for the model where particles have a G(1 − p)-distributed lifespan, are investigated in [2,8,12,16].…”

“…Shape theorems on Z d can be found in [2,3]. Phase transitions for the model where particles have a G(1 − p)-distributed lifespan, are investigated in [2,8,12,16]. Recently, in [11], global survival of an asymmetric inhomogeneous random walk system on Z has been studied (in that model particles die after L steps without activation).…”