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 S d 0( t) is within an ε neighborhood of the set [ C d t], where for A⊂ R d we define [A]:=A∩ Z d . 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 d 0( t) converges weakly to a product Poisson measure of parameter one. To cite this article: A.F. Ramı́rez, V. Sidoravicius, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 821–826.