\magnification=1200
\hsize=4in
\nopagenumbers
\noindent
%
%
{\bf Jason P. Bell}
%
%
\medskip
\noindent
%
%
{\bf When Structures Are Almost Surely Connected}
%
%
\vskip 5mm
\noindent
%
%
%
%
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.
Wright 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 Compton; moreover, we show
that $\liminf_{n\rightarrow\infty} C_n/A_n$ can assume any value between
$0$ and $1$.
\bye