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**1 - 10**of**10**### Point-to-point distance in first passage percolation on (tree)×Z, arXiv:1310.4018

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### Université de Paris-Est Marne-la-Vallée

"... Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de ..."

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Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de

### THÈSE Pour l’obtention du grade de DOCTEUR DE L’UNIVERSITÉ PARIS-EST ÉCOLE DOCTORALE MATHEMATIQUES ET SCIENCES ET TECHNOLOGIE DE L’INFORMATION ET DE LA COMMUNICATION DISCIPLINE: MATHÉMATIQUES

, 2013

"... Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de ..."

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Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement Directeur de thèse: Djalil CHAFAÏ Soutenue le vendredi 14 juin 2013 Devant le jury composé de

### Routing on trees

, 2014

"... We consider three dierent schemes for signal routing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with i.i.d. weights representing the strength of the transceivers. The edges of the tree are also equipped with i.i.d. weights, repr ..."

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We consider three dierent schemes for signal routing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with i.i.d. weights representing the strength of the transceivers. The edges of the tree are also equipped with i.i.d. weights, representing the costs for passing the edges. For each one of our schemes, we derive sharp conditions on the distributions of the vertex weights and the edge weights that determine when the root can transmit a signal over arbitrarily large distances.

### EXTINCTION WINDOW OF MEAN FIELD BRANCHING ANNIHILATING RANDOM WALK

"... Abstract. We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring the particles are annihilated. This is a non-monotone model, which mak ..."

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Abstract. We study a model of growing population that competes for resources. At each time step, all existing particles reproduce and the offspring randomly move to neighboring sites. Then at any site with more than one offspring the particles are annihilated. This is a non-monotone model, which makes the analysis more difficult. We consider the extinction window of this model in the finite mean-field case, where there are n sites but movement is allowed to any site (the complete graph). We show that although the system survives for exponential time, the extinction window is logarithmic. 1.