Boolean mean field spin glass model: rigorous results

Abstract

Spin glasses have played a fundamental role in statistical mechanics field. Purpose of this work is to analyze a variation on theme of the mean field case of them, when the Ising spins are replaced to Boolean ones, i.e. {0,1} possible values. This may be useful to continue building a solid bridge between statical mechanics of spin glasses and Machine Learning techniques. We have drawn a detailed framework of this model: we have applied Guerra and Toninelli's approach to prove the existence of the thermodynamic quenched statistical pressure for this model recovering its expression using Guerra's interpolation. Specifically, we have supposed Replica Symmetric assumption and first step of Replica Symmetry Breaking approximation for the probability distribution of the order parameter of the model. Then, we analyze the stability of the resolution in both assumptions via de Almeida-Thouless line, proving that the Replica Symmetric one well describes the model apart for small values of temperature, when the Replica Symmetry Breaking is better. All the theoretical parts are supported by numerical techniques that demonstrate perfect consistency with the analytical results.