On the Pure States of the Replica Symmetry Breaking ansatz

Abstract

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 $G:{\Omega }^n\to \left[0,1\right]$ a Gibbs measure on it of Hamiltonian $\mathcal{H}$ (also non random). We interpret the pure states of a model $\left({\Omega }^n,\mu \right)$ as disjoint subsets ${\Omega }^n$ such that the conditional measures behaves like product measures as in usual mean field approximations. Starting from such definition we try to reinterpret the RSB scheme and define an approximated probability measure. We then apply our results to the Sherrington-Kirkpatrick model to obtain the Parisi formula.

This paper has been written in the period 2015-2016 within the PTCC project (Coja-Oghlan) funded by the European Research Council. 19 pages. arXiv admin note: text overlap with arXiv:1610.03941