The full replica symmetry breaking in the Ising spin glass on random regular graph
This work focuses on the extension of the Parisi full replica symmetry breaking solution to the Ising spin glass on a random regular graph. We propose a new martingale approach, that overcomes the limits of the Parisi-Mézard cavity method, providing a well-defined formulation of the full replica symmetry breaking problem in random regular graphs. We obtain a variational free energy functional, defined by the sum of two variational functionals (auxiliary variational functionals), that is an extension of the Parisi functional of the Sherrington-Kirkpatrick model. We study the properties of the two variational functionals in details, providing representation through the solution of a proper backward stochastic differential equation, that generalizes the Parisi partial differential equation. Finally, we define the order parameters of the system and get a set of self-consistency equations for the order parameters and free energy.