On the concentration of the number of solutions of random satisfiability formulas. Issue 3 (12th April 2013) - Abbe, Emmanuel
ABSTRACT 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)1+δ) for some δ>0. We show that (possibly excluding a countable set of "exceptional" α's) the number of solutions concentrates, i.e., there exists a non‐random function α↦ϕs(α) such that, for any ε>0, we have Z(F)∈[2n(ϕs−ε), 2n(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. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 362–382, 2014