Threshold values of Random K-SAT from the cavity method

Using the cavity equations of \cite{mezard:parisi:zecchina:02,mezard:zecchina:02}, we derive the various threshold values for the number of clauses per variable of the random $K$-satisfiability problem, generalizing the previous results to $K \ge 4$. 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.