Percolation of Finite Clusters and Shielded Paths

Autor: Bounghun Bock, Vladas Sidoravicius, Charles M. Newman, Michael Damron
Rok vydání: 2020
Předmět:
Zdroj: Journal of Statistical Physics. 179:789-807
ISSN: 1572-9613
0022-4715
DOI: 10.1007/s10955-020-02558-4
Popis: In independent bond percolation on $${\mathbb {Z}}^d$$ with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite connected component? Grimmett-Holroyd-Kozma used the triangle condition to show that for $$d \ge 19$$, the set of such p contains values strictly larger than the percolation threshold $$p_c$$. With the work of Fitzner-van der Hofstad, this has been reduced to $$d \ge 11$$. We improve this result by showing that for $$d \ge 10$$ and some $$p>p_c$$, there are infinite paths consisting of “shielded” vertices—vertices all whose adjacent edges are closed—which must be in the complement of the infinite cluster. Using values of $$p_c$$ obtained from computer simulations, this bound can be reduced to $$d \ge 7$$. Our methods are elementary and do not require the triangle condition.
Databáze: OpenAIRE
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