On the concentration of the number of solutions of random satisfiability formulas

Let Z(F ) be the number of solutions of a random k-satisfiability formula F with n variables and clause density α. Assume that the probability that F is unsatisfiable is O(1/ log(n)) for some δ > 0. We show that (possibly excluding a countable set of ‘exceptional’ α’s) the normalized logarithm of the number of solutions concentrates, i.e., there exists a non-random function α 7→ φs(α) such that, for any ε > 0, we have (1/n) logZ(F ) ∈ [φs − ε, φs + ε] with high probability. In particular, the assumption holds for all α < 1, which proves the above concentration claim in the whole satisfiability regime of random 2-SAT. We also extend these results to a broad class of constraint satisfaction problems.