Replica Symmetry Breaking without replicas

Simone Franchini

Published 2016-10-13Version 1

We discuss the concept of pure state of the Replica Symmetry Breaking ansatz in finite and infinite spin systems without averaging on the disorder, nor using replicas. Consider a system of $n$ spins $\sigma\in\Omega^{n}$ with the usual set $\Omega=\left\{ -1,1\right\}$ of inner states and let $\mu:\,\Omega^{n}\rightarrow\left[0,1\right]$ a probability measure on it (also random). We interpret the pure states as components of a nontrivial partition of $\Omega^{n}$ such that the measure conditioned to each component behaves like a product measure. Starting from such definition we are able to derive a very general variational principle. Then we reinterpret the assumptions of the RSB scheme to define a sequence of approximated probability measure and, finally, we apply our results to the Sherrington-Kirkpatrick model to obtain the Parisi formula.