Statistical mechanics of the random K-satisfiability model

The random K-satisfiability problem. consisting in verifying the existence of an assignment of N Boolean variables that satisfy a set of M = alpha N random logical clauses containing K variables each, is studied using the replica symmetric framework of diluted disordered systems. We present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest K = 2, K greater than or equal to 3, and K much greater than 1. The calculation of the number of solutions, which allowed us [Phys. Rev. Lett. 76, 3881 (1996)] to predict a first order jump at the threshold where the Boolean expressions become unsatisfiable with probability one, is thoroughly displayed. In the case K=2, the (rigorously known) critical value (alpha=1) of the number of clauses per Boolean variable is recovered while for K greater than or equal to 3 we show that the system exhibits a replica symmetry breaking transition. The annealed approximation is proven to be exact for large K.