Enlargement of subgraphs of infinite graphs by Bernoulli percolation

Abstract

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 [24] without proofs. This paper gives full details of them.

Unions of random walk and percolation on infinite graphs

“…Consider Bernoulli bond percolation on an infinite connected graph G. Assume that each edge of G is open with probability p ∈ [0, 1] and closed with probability 1 − p. It seems natural to consider the following informal question: if we add Bernoulli percolation on G to a subgraph H of G, then how much does H change? By this motivation, the author [Okam17] proposed a model in which a configuration of Bernoulli percolation on an infinite connected graph G is added to a (deterministic or random) subgraph H independently, and then, asked whether the probability that a property P of H remains to be satisfied for the enlargement of H is less than 1, as p increases. If H is a single vertex of G and P is the property that the graph has an infinite number of vertices, we obtain the definition of Hammersley's critical probability.…”

“…If H is a single vertex of G and P is the property that the graph has an infinite number of vertices, we obtain the definition of Hammersley's critical probability. In [Okam17], an important example of such a subgraph H and a property P is the case that H is the trace of the simple random walk {S n } n on G and P is that the (enlarged) graph is recurrent, that is, the simple random walk on the graph is recurrent.…”

“…In this paper, we terminate the simple random walk on an infinite connected simple graph at a time n, and consider asymptotics for the number of vertices of the enlargement of the trace until the time n by subcritical Bernoulli percolation on the same graph. The main focus of [Okam17] is the case that H is infinite. On the other hand, we focus on the case that H = H n is finite, and Furthermore depends on the time n. We denote by U n the number of the vertices of the union of the simple random walk until time n and Bernoulli percolation (see Definition 1.1).…”

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

“…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 [24] without proofs. This paper gives full details of them.…”

“…Most results of this paper are announced in [24] without proofs. This paper gives full details of them.…”