Abstract. In an influential article, Papadimitriou [ On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1991, pp. 163–169] proved that a local search algorithm called WalkSAT finds a satisfying assignment of a satisfiable 2-CNF with [Formula: see text] variables in [Formula: see text] expected time. Variants of the WalkSAT algorithm have become a mainstay of practical SAT solving (see, e.g., [Hoos and Stützle, J. Autom. Reason., 24 (2000), pp. 421–481]). In the present article, we analyze the expected running time of WalkSAT on random 2-SAT instances. Answering a question raised by Alekhnovich and Ben-Sasson [ SIAM J. Comput., 36 (2007) pp. 1248–1263], we show that WalkSAT runs in linear expected time for all clause/variable densities up to the random 2-SAT satisfiability threshold.

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order √n, with n the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are bounded throughout all or most of the satisfiable regime.