Long-range frustration in finite connectivity spin glasses: a mean-field theory and its application to the random K-satisfiability problem

Abstract

A mean-field theory of long-range frustration is constructed for spin glass systems with quenched randomness of vertex–vertex connections and of spin–spin coupling strengths. This theory is applied to a spin glass model of the random K-satisfiability (K-SAT) problem (K=2 or K=3). The satisfiability transition in a random 2-SAT formula occurs when the clauses-to-variables ratio α approaches αc(2)=1. However, long-range frustration among unfrozen variable nodes builds up only when α>αR(2)=4.4588. For the random 3-SAT problem, we find a long-range frustrated mean-field solution when α>αR(3)=4.1897. The long-range frustration order parameter R of this solution jumps from zero to a finite positive value at αR(3), while the energy density increases only gradually from zero as a function of α. The SAT–UNSAT transition point of this solution is lower than the value of αc(3)=4.267 obtained by the survey propagation algorithm. Two possible reasons for this discrepancy are suggested. The zero-temperature phase diagram of the ±J Viana–Bray model is also determined, which is identical to that of the random 2-SAT problem. The predicted phase transition between a non-frustrated and a long-range frustrated spin glass phase might also be observable in real materials at a finite temperature.