Enlargement of subgraphs of infinite graphs by Bernoulli percolation

Kazuki Okamura

Published 2015-06-19Version 1

We consider changes of properties of subgraphs of an infinite graph if we enlarge the subgraphs by adding Bernoulli percolation on the infinite graph to them. We give a pair of an infinite graph, a subgraph of it, and, a property. Then, we can define two "critical probabilities" of the pair, in the same manner as the (ordinal) critical probability is defined. We focus on the following cases that a property is being a transient subgraph, having finitely many or no cut points, being a recurrent subset, and, being connected. We compare the two critical probabilities with the critical probability. Our results depend heavily on a choice of a pair of an infinite graph, a subgraph of it, and, a property.