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

TL;DR

This paper investigates a modified percolation process on the square lattice where clusters freeze upon reaching a certain diameter, exploring the behavior as the parameter N increases and providing partial answers to related probabilistic questions.

Contribution

It introduces and analyzes a new cluster freezing process on the square lattice based on diameter, addressing questions about its asymptotic behavior as N grows.

Findings

Partial understanding of the probability that a given cluster freezes as N increases

Insights into the size distribution of final clusters in the process

Discussion of anomalous behaviors as the parameter N tends to infinity

Abstract

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 tends to infinity. For instance, if one looks at the cluster of a given vertex, does, as N tends to infinity, 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.