Sharp thresholds and the partition function

Abstract

Let Φ be a random k-CNF formula over n variables with clauses and let Z(Φ) be the number of satisfying assignments. Assume that k is sufficiently large and that r ≤ (1 - ok(1))2k ln(k)/k, where o k(1) denotes a certain function that tends to 0 as k gets large. We prove that in this case, Φ is satisfiable with probability 1 - O(1/n). Together with a recent result of Abbe and Montanari [arXiv:1006.3786], this implies that for such k,r the limit exists. The existence of this limit is related to the existence of a sharp threshold for satisfiability. © Published under licence by IOP Publishing Ltd.