Abstract

In this chapter we analyze some properties of the overlap probability distribution by looking at its factorization rules. It has been now understood that the disputed features of the spin glass phase are, apart from the triviality issue of the single overlap distribution, related to the structure of the joint overlap distribution. For example, a centered Gaussian family distribution is completely identified by its covariance, thanks to the Wick theorem factorization law. In the spin glass phase, the factorization structure first appeared within the Parisi replica symmetry breaking solution of the Sherrington–Kirkpatrick model. In this chapter we show that some of those properties can indeed be recovered by using a stability argument which amounts to proving how the equilibrium state is left unchanged by small perturbations. In turn, this is equivalent to controlling the size of the fluctuations for suitable thermodynamic quantities. After introducing the general method we review the stochastic stability, the control of thermal fluctuations, and the graph-theoretical description of the emerging identities. We then analyze the fluctuations due to the disorder and establish the self-averaging of the random internal energy. The extension to interactions with non-zero averages and the specific analysis on the Nishimori line follows. The chapter ends with newidentities derived from the control of the fluctuations for free energy differences involving flips of the interactions.

The stability method and the structural identities

The solution of the Sherrington–Kirkpatrick model obtained via the replica approach showed that the order parameter of the theory, namely the overlap distribution, is sufficient to fully describe the quenched state.