The infinitary logic of sparse random graphs

Let L_∞ω^ω be the infinitary language obtained from the first-order language of graphs by closure under conjunctions and disjunctions of arbitrary sets of formulas, provided only finitely many distinct variables occur among the formulas. Let p(n) be the edge probability of the random graph on n vertices. Previous articles have shown that when p(n) is constant or p(n)=n^-α and α>1, then every sentence in L_∞ω^ω has probability that converges as n gets large; however, when p(n)=n^-α and α<1 is rational, then there are first-order sentences whose probability does not converge. This article completes the picture for L _∞ω^ω and random graphs with edge probability of the form n^-α. It is shown that if α is irrational, then every sentence in L_∞ω ^ω has probability that converges to 0 or 1. It is also shown that if α=1, then there are sentences an the deterministic transitive closure logic (and therefore in L_∞ω ^ω), whose probability does not converge