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

Abstract 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 Ref. [1] to more general combinatorial problems. It turns out that 1RSB is always unstable at sufficiently small clauses density α or high energy. In particular, the recent 1RSB solution to 3-SAT is unstable at zero energy for α < αm, with αm ≈ 4.153. On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB.