Enlargement of subgraphs of infinite graphs by Bernoulli percolation

TL;DR

This paper investigates how adding Bernoulli percolation edges affects properties of subgraphs within infinite graphs, defining critical probabilities and comparing them to Hammersley's critical probability.

Contribution

It introduces a framework for analyzing property changes in subgraphs under Bernoulli percolation, with detailed proofs of previously announced results.

Findings

Defined two critical probabilities related to subgraph properties.

Compared these critical probabilities with Hammersley's critical probability.

Analyzed properties like transience, recurrence, and connectivity under percolation.

Abstract

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs, and a property of the subgraphs. Then, in a manner similar to the way Hammersley's critical probability is defined, we can define two values associated with the triplet. We regard the two values as certain critical probabilities, and compare them with Hammersley's critical probability. In this paper, we focus on the following cases of a graph property: being a transient subgraph, having finitely many cut points or no cut points, being a recurrent subset, or being connected. Our results depend heavily on the choice of the triplet. Most results of this paper are announced in \cite{O16} without proofs. This paper gives full details of them.