Random spatial growth with paralyzing obstacles

van den Berg, J.; Peres, Y.; Sidoravicius, V.; Vares, M. E.

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1173-1187.

Nous étudions des modèles de croissance spatiaux qui, au temps initial, ont des sources de croissance (indiquées par la couleur verte) et des sources de substance paralysante arrêtant la croissance (indiquées en rouge). Les sources vertes augmentent et peuvent fusionner avec les autres (il n’y a pas de compétition entre elles). La substance rouge reste passive quand elle est isolée. Cependant, quand un amas vert touche la substance rouge, il est envahi immédiatement par cette dernière, il arrête de grandir et commence à agir comme la substance rouge. Dans notre modèle principal, l’espace est représenté par un graphe dont, à l’instant initial, tous les sommets sont tirés au hasard vert, rouge ou blanc (vide) et la croissance des amas verts est similaire à celle de la percolation de premier-passage. Les problèmes principaux que nous considérons sont les suivants ; est-ce que le modèle est bien défini sur un graphe infini (par exemple le treillis $d$-dimensionnel) ? Que peut-on dire de la distribution de la taille d’un amas vert juste avant qu’il soit paralysé ? Nous montrons que, si la densité initiale de sommets rouges est positive et que celle des sommets blancs est suffisamment petite, le modèle est en effet bien défini et la distribution ci-dessus mentionnée a une queue exponentielle. Nous conjecturons que ce résultat est vrai dès que la densité initiale des rouges est positive. Ce travail mène également à une relation entre la percolation d’invasion et la percolation de Bernouilli critique qui semble être d’intérêt indépendement.

We study models of spatial growth processes where initially there are sources of growth (indicated by the colour green) and sources of a growth-stopping (paralyzing) substance (indicated by red). The green sources expand and may merge with others (there is no ‘inter-green’ competition). The red substance remains passive as long as it is isolated. However, when a green cluster comes in touch with the red substance, it is immediately invaded by the latter, stops growing and starts to act as a red substance itself. Our main model space is represented by a graph, of which initially each vertex is randomly green, red or white (vacant), and the growth of the green clusters is similar to that in first-passage percolation. The main issues we investigate are whether the model is well defined on an infinite graph (e.g. the $d$-dimensional cubic lattice), and what can be said about the distribution of the size of a green cluster just before it is paralyzed. We show that, if the initial density of red vertices is positive, and that of white vertices is sufficiently small, the model is indeed well defined and the above distribution has an exponential tail. In fact, we believe this to be true whenever the initial density of red is positive. This research also led to a relation between invasion percolation and critical Bernoulli percolation which seems to be of independent interest.

Classification : 60K35, 60K37, 82B43
Mots clés : growth process, percolation, invasion percolation

[1] D. J. Aldous. The percolation process on a tree where infinite clusters are frozen. Proc. Camb. Phil. Soc. 128 (2000) 465-477. | MR | Zbl

[2] K. S. Alexander. Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 (1995) 87-104. | MR | Zbl

[3] I. Benjamini and O. Schramm. Private communication, 1999.

[4] J. Van Den Berg and B. Tóth. A signal-recovery system: asymptotic properties, and construction of an infinite-volume process. Stochastic Process. Appl. 96 (2001) 177-190. | MR | Zbl

[5] J. Van Den Berg, A. Járai and B. Vágvölgyi. The size of a pond in 2D invasion percolation. Electron. Comm. Probab. 12 (2007) 411-420. | MR | Zbl

[6] J. T. Chayes, L. Chayes and C. M. Newman. Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. 15 (1987) 1272-1287. | MR | Zbl

[7] M. Dürre. Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron J. Probab. 11 n. 21, (2006) 513-539 (electronic). | MR | Zbl

[8] G. R. Grimmett. Percolation, 2nd edition. Springer, 1999. | MR

[9] O. Häggström and R. Meester. Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 (1996) 295-315. | MR | Zbl

[10] O. Häggström, Y. Peres and R. H. Schonmann. Percolation on transitive graphs as a coalescent process: relentless merging followed by simultaneous uniqueness. In Perplexing Problems in Probability (M. Bramson and R. Durrett, Eds) 44 69-90. Birkhäuser, Boston, 1999. | MR | Zbl

[11] A. Járai. Private communication, 1999.

[12] A. Járai. Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys. 236 (2003) 311-334. | MR | Zbl

[13] H. Kesten. Analyticity properties and power law estimates in percolation theory. J. Statist. Phys. 25 (1981) 717-756. | MR | Zbl

[14] H. Kesten. Scaling relations for 2D percolation. Comm. Math. Phys. 109 (1987) 109-156. | MR | Zbl

[15] R. Lyons, Y. Peres. Probability on trees and networks. Available at http://mypage.iu.edu/~rdlyons/. | Zbl

[16] R. Lyons, Y. Peres and O. Schramm. Minimal spanning forests. Ann. Probab. 34 (2006) 1665-1692. | MR | Zbl

[17] D. L. Stein and C. M. Newman. Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E 51 (1995) 5228-5238.

[18] D. Wilkinson and J. F. Willemsen. Invasion percolation: a new form of percolation theory. J. Phys. A 16 (1983) 3365-3376. | MR