✍ Scribed by Alejandro F. Ramı́ rez; Vladas Sidoravicius
No coin nor oath required. For personal study only.
We study a continuous time growth process on Z d (d 1) associated to the following interacting particle system: initially there is only one simple symmetric continuous time random walk of total jump rate one located at the origin; then, whenever a random walk visits a site still unvisited by any other random walk, it creates a new independent random walk starting from that site. Let us call P d the law of such a process and S 0 d (t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set C d ⊂ R d , such that for every ε > 0, P d -a.s. eventually in t, the set
Moreover, for d large enough, the set C d is not a ball under the Euclidean norm. We also show that the empirical density of particles within S 0 d (t) converges weakly to a product Poisson measure of parameter one.
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