Replica symmetry breaking in weak connectivity systems

Abstract

The authors propose a generalisation of Parisi's replica symmetry breaking scheme (1979) to spin glasses (or optimisation problems) described by a set of order parameters Q(r)alpha 1. . . alpha r, or equivalently by a global order parameter G( sigma alpha ), rather than by Q(r)alpha 1 alpha 2 alone. They study the particular case of Derrida's p-spin model (1981) in the very dilute limit. The model is solved exactly in the large p limit and a freezing transition occurs. Using replicas and a single step of their replica symmetry breaking ansatz they recover the exact result. They discuss the implications for the global order parameter G( sigma alpha ) when more than one step in the replica symmetry breaking process is taken.