Comptes Rendus
no. 10
Probability Theory
Ergodic properties of geometrical crystallization processes

Presented by: Paul Deheuvels

Youri Davydov1; Aude Illig2
1 Université des sciences et technologies de Lille, Laboratoire Paul-Painlevé, UMR 8524, 59655 Villeneuve d'Ascq cedex, France
2 Université Versailles Saint-Quentin, Laboratoire de mathématiques de Versailles, UMR 8100, 45, avenue des États-Unis, 78035 Versailles, France
Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 583-586.

We consider a birth and growth process with germs being born according to a Poisson point process whose intensity measure is invariant under translations in space. The germs can be born in unoccupied space and then start growing until they occupy the available space. In this general framework, the crystallization process can be characterized by a random field which, for any point in the state space, assigns the first time at which this point is reached by a crystal. Under general conditions on the growth speed and geometrical shape of free crystals, we prove that the random field is mixing in the sense of ergodic theory, and we also obtain estimates for the absolute regularity coefficient.

Nous nous intéressons à la cristallisation d'un domaine par des germes apparaissant selon un processus ponctuel de Poisson d'intensité invariante par translation spatiale. Les germes se fixent uniquement en zone libre et se mettent ensuite à croître pour former des cristaux qui occupent progressivement l'espace. Ce procédé peut être décrit par le champ aléatoire donnant en tout point de l'espace le premier instant de recouvrement par un cristal. Nous démontrons sous des hypothèses générales sur la vitesse de croissance et la forme des cristaux libres que le processus est mélangeant au sens de la théorie ergodique et obtenons des estimations du coefficient de régularité absolue.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.10.017

Youri Davydov 1; Aude Illig 2

1 Université des sciences et technologies de Lille, Laboratoire Paul-Painlevé, UMR 8524, 59655 Villeneuve d'Ascq cedex, France
2 Université Versailles Saint-Quentin, Laboratoire de mathématiques de Versailles, UMR 8100, 45, avenue des États-Unis, 78035 Versailles, France 
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Youri Davydov; Aude Illig. Ergodic properties of geometrical crystallization processes. Comptes Rendus. Mathématique, Volume 345 (2007) no. 10, pp. 583-586. doi : 10.1016/j.crma.2007.10.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.017/

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