Threshold values of random K-SAT from the cavity method

Using the cavity equations of Mezard, Parisi, and Zecchina [Science 297 (2002), 812; Mezard 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 any K, 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. (c) 2005 Wiley Periodicals, Inc.