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{\bf Katherine St.~John}\medskip\noindent
{\bf Limit Probabilities for Random Sparse Bit Strings}
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	Let $n$ be a positive integer, $c$ a real positive constant, and 
	$p(n) = c/n$.  Let $U_{n,p}$ be the random unary
	predicate under the linear order,
	and $S_c$ the almost sure theory of 
$U_{n,{c\over n}}$.
	We show that for every first-order sentence $\phi$:
	$$
		f_{\phi}(c) = \lim_{n\rightarrow\infty}{\Pr}[U_{n,{c\over n}}
{ has\ property\ } \phi]
	$$
	is an infinitely differentiable function.  
	Further, let $S = \bigcap_c S_c$ be the set of all sentences
	that are true in every almost sure theory.  Then, for every $c>0$,
	$S_c = S$.
\end