  We extend the convergence law for sparse random graphs proven by Lynch to
arbitrary relational languages. We consider a finite relational vocabulary
$\sigma$ and a first order theory $T$ for $\sigma$ composed of symmetry and
anti-reflexivity axioms. We define a binomial random model of finite
$\sigma$-structures that satisfy $T$ and show that first order properties have
well defined asymptotic probabilities when the expected number of tuples
satisfying each relation in $\sigma$ is linear. It is also shown that these
limit probabilities are well-behaved with respect to several parameters that
represent the density of tuples in each relation $R$ in the vocabulary
$\sigma$. An application of these results to the problem of random Boolean
satisfiability is presented. We show that in a random $k$-CNF formula on $n$
variables, where each possible clause occurs with probability $\sim c/n^{k-1}$,
independently any first order property of $k$-CNF formulas that implies
unsatisfiability does almost surely not hold as $n$ tends to infinity.
