TY - JOUR

T1 - A percolation process on the square lattice where large finite clusters are frozen

AU - van den Berg, J.

AU - de Lima, B.N.B.

AU - Nolin, P.

PY - 2012

Y1 - 2012

N2 - In (Aldous, Math. Proc. Cambridge Philos. Soc. 128 (2000), 465-477), Aldous constructed a growth process for the binary tree where clusters freeze as soon as they become infinite. It was pointed out by Benjamini and Schramm that such a process does not exist for the square lattice. This motivated us to investigate the modified process on the square lattice, where clusters freeze as soon as they have diameter larger than or equal to N, the parameter of the model. The non-existence result, mentioned above, raises the question if the N- parameter model shows some 'anomalous' behaviour as N →∞. For instance, if one looks at the cluster of a given vertex, does, as N →∞, the probability that it eventually freezes go to 1? Does this probability go to 0? More generally, what can be said about the size of a final cluster? We give a partial answer to some of such questions. © 2011 Wiley Periodicals, Inc.

AB - In (Aldous, Math. Proc. Cambridge Philos. Soc. 128 (2000), 465-477), Aldous constructed a growth process for the binary tree where clusters freeze as soon as they become infinite. It was pointed out by Benjamini and Schramm that such a process does not exist for the square lattice. This motivated us to investigate the modified process on the square lattice, where clusters freeze as soon as they have diameter larger than or equal to N, the parameter of the model. The non-existence result, mentioned above, raises the question if the N- parameter model shows some 'anomalous' behaviour as N →∞. For instance, if one looks at the cluster of a given vertex, does, as N →∞, the probability that it eventually freezes go to 1? Does this probability go to 0? More generally, what can be said about the size of a final cluster? We give a partial answer to some of such questions. © 2011 Wiley Periodicals, Inc.

U2 - 10.1002/rsa.20375

DO - 10.1002/rsa.20375

M3 - Article

VL - 40

SP - 220

EP - 226

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

ER -