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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.

Probabilities of first-order sentences on sparse random relational structures: An application to definability on random CNF formulas