Instability of one-step replica-symmetry-broken phase in satisfiability problems

We reconsider the one-step replica-symmetry-breaking (1RSB) solutions of two random combinatorial problems: k-XORSAT and k-SAT. We present a general method for establishing the stability of these solutions with respect to further steps of replica-symmetry breaking. Our approach extends the ideas of Montanari and Ricci-Tersenghi (2003 Eur. Phys. J. B 33 339) to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clause density a or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for α < α _m, with α_m < 4.153. On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB.