  This thesis focus 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-M\'ezard
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 are an extension of the Parisi
functional of the Sherrington-Kirkpatrick model. We study the properties of the
two variational functionals in detailed, providing representation through the
solution of a proper backward stochastic differential equation, that generalize
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.
