 
Summary: Percolation on finite graphs and isoperimetric inequalities
Noga Alon
Itai Benjamini
Alan Stacey
Abstract
Consider a uniform expanders family Gn with a uniform bound on the degrees. It is
shown that for any p and c > 0, a random subgraph of Gn obtained by retaining each
edge, randomly and independently, with probability p, will have at most one cluster of
size at least cGn, with probability going to one, uniformly in p and the size of the
expander. The method from Ajtai, Koml´os and Szemer´edi [1] is applied to obtain some
new results about the critical probability for the emergence of a giant component in
random subgraphs of finite regular expanding graphs of high girth, as well as a simple
proof of a result of Kesten [16] about the critical probability for bond percolation in high
dimensions. Several problems and conjectures regarding percolation on finite transitive
graphs are presented.
1 Introduction
In this paper we primarily consider percolation on finite graphs, and in particular the existence
and uniqueness of large components, typically meaning components whose size is proportional
to the number of vertices in the graph. Our main results in this context apply to expanders,
which are graphs satisfying a particular isoperimetric inequality, although we conjecture that
