When Structures Are Almost Surely Connected

Let An denote the number of objects of some type of “size” n, and let Cn denote the number of these objects which are connected. It is often the case that there is a relation between a generating function of the Cn’s and a generating function of the An’s. Wright showed that if limn→∞Cn/An = 1, then the radius of convergence of these generating functions must be zero. In this paper we prove that if the radius of convergence of the generating functions is zero, then lim supn→∞Cn/An = 1, proving a conjecture of Compton; moreover, we show that lim infn→∞Cn/An can assume any value between 0 and 1.