Spin glass polynomial identities from entropic constraints

TL;DR

This paper develops a novel entropic approach to derive polynomial identities in diluted spin glasses, extending previous energy-based methods and revealing more identities under certain conditions, with implications for understanding spin glass behavior.

Contribution

It introduces an entropic constraints method to derive polynomial identities in diluted spin glasses, expanding the theoretical framework beyond energy-based approaches.

Findings

Entropic constraints produce more identities than energy-based methods in diluted spin glasses.

On fully connected topologies, entropic and energy-based identities coincide, matching known results.

The approach provides a new perspective on overlap constraints and stochastic stability in spin glasses.

Abstract

The core idea of stochastic stability is that thermodynamic observables must be robust under small (random) perturbations of the quenched Gibbs measure. Combining this idea with the cavity field technique, which aims to measure the free energy increment under addition of a spin to the system, we sketch how to write a stochastic stability approach to diluted mean field spin glasses which explicitly gives overlap constraints as the outcome. We then show that, under minimal mathematical assumptions and for gauge invariant systems (namely those with even Ising interactions), it is possible to "reverse" the idea of stochastic stability and use it to derive a broad class of constraints on the unperturbed quenched Gibbs measure. This paper extends a previous study where we showed how to derive (linear) polynomial identities from the "energy" contribution to the free energy, while here we focus on the consequences of "entropic" constraints. Interestingly, in diluted spin glasses, the entropic approach generates more identities than those found by the energy route or other techniques. The two sets of identities become identical on a fully connected topology, where they reduce to the ones derived by Aizenman and Contucci.