When structures are almost surely connected (Q1572760)

Summary: Let \(A_n\) denote the number of objects of some type of ``size'' \(n\), and let \(C_n\) 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 \(C_n\)'s and a generating function of the \(A_n\)'s. \textit{E. M. Wright} [Proc. Lond. Math. Soc., III. Ser. 17, 296-304 and 547-552 (1967; Zbl 0147.31901) and J. Lond. Math. Soc. 43, 720-724 (1968; Zbl 0159.25501)] showed that if \(\lim_{n\rightarrow\infty} C_n/A_n =1\), 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 \(\limsup_{n\rightarrow \infty} C_n/A_n =1\), proving a conjecture of \textit{K. J. Compton} [Discrete Math. 66, 59-77 (1987; Zbl 0649.05012)]; moreover, we show that \(\liminf_{n\rightarrow\infty} C_n/A_n\) can assume any value between \(0\) and \(1\).