We study a continuous time growth process on (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 Pd the law of such a process and S0d(t) the set of sites, visited by all walks by time t. We prove that there exists a bounded, non-empty, convex set , such that for every ε>0, Pd-a.s. eventually in t, the set Sd0(t) is within an ε neighborhood of the set [Cdt], where for we define . Moreover, for d large enough, the set Cd is not a ball under the Euclidean norm. We also show that the empirical density of particles within Sd0(t) converges weakly to a product Poisson measure of parameter one.
Nous étudions un modèle de croissance à temps continu sur (d⩾1) associé au système de particules en interaction suivant : initiallement il y a seulement une marche aléatoire à temps continu simple, symétrique, à taux un et située à l'origine ; ensuite, aussitôt qu'une marche aléatoire visite un site jamais encore visité par aucune autre marche, une nouvelle marche est créée, partant de ce site et indépendante des autres. Nous notons Pd la loi d'un tel processus et S0d(t) l'ensemble des sites déjà visités à l'instant t. Nous prouvons qu'il existe un ensemble convexe, non-vide et borné tel que, pour tout ε>0, Pd-p.s. et pour t assez grand, l'ensemble Sd0(t) soit inclus dans un ε-voisinage de [Cdt], où l'on a défini, pour , . En outre, pour d assez grand, l'ensemble Cd n'est pas une boule pour la norme euclidienne. Enfin, nous montrons que la densité empirique de particules à l'intérieur de Sd0(t) converge faiblement vers un produit de lois de Poisson de paramétre un.
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Alejandro F. Ramı́ rez 1; Vladas Sidoravicius 2
@article{CRMATH_2002__335_10_821_0, author = {Alejandro F. Ram{\i}́ rez and Vladas Sidoravicius}, title = {Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walks}, journal = {Comptes Rendus. Math\'ematique}, pages = {821--826}, publisher = {Elsevier}, volume = {335}, number = {10}, year = {2002}, doi = {10.1016/S1631-073X(02)02568-2}, language = {en}, }
TY - JOUR AU - Alejandro F. Ramı́ rez AU - Vladas Sidoravicius TI - Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walks JO - Comptes Rendus. Mathématique PY - 2002 SP - 821 EP - 826 VL - 335 IS - 10 PB - Elsevier DO - 10.1016/S1631-073X(02)02568-2 LA - en ID - CRMATH_2002__335_10_821_0 ER -
%0 Journal Article %A Alejandro F. Ramı́ rez %A Vladas Sidoravicius %T Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walks %J Comptes Rendus. Mathématique %D 2002 %P 821-826 %V 335 %N 10 %I Elsevier %R 10.1016/S1631-073X(02)02568-2 %G en %F CRMATH_2002__335_10_821_0
Alejandro F. Ramı́ rez; Vladas Sidoravicius. Asymptotic behavior of a stochastic growth process associated with a system of interacting branching random walks. Comptes Rendus. Mathématique, Volume 335 (2002) no. 10, pp. 821-826. doi : 10.1016/S1631-073X(02)02568-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02568-2/
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