Threshold values of Random K-SAT from the cavity method

Using the cavity equations of Mézard, Parisi, and Zecchina [Science 297 (2002), 812; Mézard and Zecchina, Phys Rev E 66 (2002), 056126] we derive the various threshold values for the number of clauses per variable of the random K-satisfiability problem, generalizing the previous results to K ≥ 4. We also give an analytic solution of the equations, and some closed expressions for these thresholds, in an expansion around large K . The stability of the solution is also computed. For anyK , the satisfiability threshold is found to be in the stable region of the solution, which adds further credit to the conjecture that this computation gives the exact satisfiability threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 28, 000–000, 2006