Random spatial growth with paralyzing obstacles

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

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 6, page 1173-1187
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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van den Berg, J., et al. "Random spatial growth with paralyzing obstacles." Annales de l'I.H.P. Probabilités et statistiques 44.6 (2008): 1173-1187. <http://eudml.org/doc/78008>.

@article{vandenBerg2008,
abstract = {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.},
author = {van den Berg, J., Peres, Y., Sidoravicius, V., Vares, M. E.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {growth process; percolation; invasion percolation},
language = {eng},
number = {6},
pages = {1173-1187},
publisher = {Gauthier-Villars},
title = {Random spatial growth with paralyzing obstacles},
url = {http://eudml.org/doc/78008},
volume = {44},
year = {2008},
}

TY - JOUR
AU - van den Berg, J.
AU - Peres, Y.
AU - Sidoravicius, V.
AU - Vares, M. E.
TI - Random spatial growth with paralyzing obstacles
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 6
SP - 1173
EP - 1187
AB - 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.
LA - eng
KW - growth process; percolation; invasion percolation
UR - http://eudml.org/doc/78008
ER -

References

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  1. [1] D. J. Aldous. The percolation process on a tree where infinite clusters are frozen. Proc. Camb. Phil. Soc. 128 (2000) 465–477. Zbl0961.60096MR1744108
  2. [2] K. S. Alexander. Percolation and minimal spanning forests in infinite graphs. Ann. Probab. 23 (1995) 87–104. Zbl0827.60079MR1330762
  3. [3] I. Benjamini and O. Schramm. Private communication, 1999. 
  4. [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. Zbl1058.60093MR1865354
  5. [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. Zbl1128.60087MR2350578
  6. [6] J. T. Chayes, L. Chayes and C. M. Newman. Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. 15 (1987) 1272–1287. Zbl0627.60099MR905331
  7. [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). Zbl1109.60081MR2242654
  8. [8] G. R. Grimmett. Percolation, 2nd edition. Springer, 1999. MR1707339
  9. [9] O. Häggström and R. Meester. Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9 (1996) 295–315. Zbl0866.60088MR1606845
  10. [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. Zbl0948.60098MR1703125
  11. [11] A. Járai. Private communication, 1999. 
  12. [12] A. Járai. Invasion percolation and the incipient infinite cluster in 2D. Comm. Math. Phys. 236 (2003) 311–334. Zbl1041.82020MR1981994
  13. [13] H. Kesten. Analyticity properties and power law estimates in percolation theory. J. Statist. Phys. 25 (1981) 717–756. Zbl0512.60095MR633715
  14. [14] H. Kesten. Scaling relations for 2D percolation. Comm. Math. Phys. 109 (1987) 109–156. Zbl0616.60099MR879034
  15. [15] R. Lyons, Y. Peres. Probability on trees and networks. Available at http://mypage.iu.edu/~rdlyons/. 
  16. [16] R. Lyons, Y. Peres and O. Schramm. Minimal spanning forests. Ann. Probab. 34 (2006) 1665–1692. Zbl1142.60065MR2271476
  17. [17] D. L. Stein and C. M. Newman. Broken ergodicity and the geometry of rugged landscapes. Phys. Rev. E 51 (1995) 5228–5238. 
  18. [18] D. Wilkinson and J. F. Willemsen. Invasion percolation: a new form of percolation theory. J. Phys. A 16 (1983) 3365–3376. MR725616

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