On one-step replica symmetry breaking in the Edwards-Anderson spin glass model

We consider a one-step replica symmetry breaking description of the Edwards-Anderson spin glass model in 2D. The ingredients of this description are a Kikuchi approximation to the free energy and a second-level statistical model built on the extremal points of the Kikuchi approximation, which are also fixed points of a Generalized Belief Propagation (GBP) scheme. We show that a generalized free energy can be constructed where these extremal points are exponentially weighted by their Kikuchi free energy and a Parisi parameter $y$, and that the Kikuchi approximation of this generalized free energy leads to second-level, one-step replica symmetry breaking (1RSB), GBP equations. We then proceed analogously to Bethe approximation case for tree-like graphs, where it has been shown that 1RSB Belief Propagation equations admit a Survey Propagation solution. We discuss when and how the one-step-replica symmetry breaking GBP equations that we obtain also allow a simpler class of solutions which can be interpreted as a class of Generalized Survey Propagation equations for the single instance graph case.