Notes on the polynomial identities in random overlap structures

TL;DR

This paper reviews random overlap structures in spin glasses, derives polynomial identities (AC relations) at all orders, and explores their implications for energy expansions and entropy's role in constraints.

Contribution

It provides a rigorous derivation of polynomial identities in ROSt frameworks for both fully connected and diluted spin glasses, highlighting recursive structures and entropy effects.

Findings

AC identities hold at all orders due to recursive polynomial structure

The set of identities is smaller than previously known

Higher-order identities involve entropy contributions

Abstract

In these notes we review first in some detail the concept of random overlap structure (ROSt) applied to fully connected and diluted spin glasses. We then sketch how to write down the general term of the expansion of the energy part from the Boltzmann ROSt (for the Sherrington-Kirkpatrick model) and the corresponding term from the RaMOSt, which is the diluted extension suitable for the Viana-Bray model. From the ROSt energy term, a set of polynomial identities (often known as Aizenman-Contucci or AC relations) is shown to hold rigorously at every order because of a recursive structure of these polynomials that we prove. We show also, however, that this set is smaller than the full set of AC identities that is already known. Furthermore, when investigating the RaMOSt energy for the diluted counterpart, at higher orders, combinations of such AC identities appear, ultimately suggesting a crucial role for the entropy in generating these constraints in spin glasses.