Ballots, queues and random graphs

This paper demonstrates how a simple ballot theorem leads, through the interjection of a queuing process, to the solution of a problem in the theory of random graphs connected with a study of polymers in chemistry. Let Γ n ( p ) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where 0 ρ n ( s ) the number of vertices in the union of all those components of Γ n ( p ) which contain at least one vertex of a given set of s vertices. This paper is concerned with the determination of the distribution of ρ n ( s ) and the limit distribution of ρ n ( s ) as n → ∞and ρ → 0 in such a way that np → a where a is a positive real number.