On the concentration of the number of solutions of random satisfiability formulas
Emmanuel A. Abbe, Andrea Montanari · Random Structures and Algorithms · 2013
ABSTRACT LetZ(F) be the number of solutions of a randomk‐satisfiability formulaFwithnvariables and clause densityα. Assume that the probability thatFis unsatisfiable is for some . 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 such that, for any , we have with high probability. In particular, the assumption holds for all , 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