Boolean SK Model

Abstract

For over half a century, statistical mechanics of spin glasses played as a paradigm to model and interpret disparate phenomena, ranging from quantitative biology to computer science. However, despite the extensive body of research in this area, there is still a notable lack of studies addressing the replacement of Ising spins with Boolean spins: as the latter play as bits in Machine Learning, this gap to fill is now mandatory. Purpose of this paper is to address this study by focusing on the mean field assumption, providing a comprehensive description of the results pertaining to these networks, referred to as the Boolean SK model due to their close relationship with the SK one. We provide a comprehensive framework for this model by employing Guerra interpolation: the thermodynamic limit, the replica symmetric and the broken replica free energy expressions are derived. Further, we inspect the onset of the replica symmetry breaking -- i.e., the de Almeida-Thouless line -- and derive Ghirlanda-Guerra fluctuations. All theoretical findings are corroborated by numerical inspections and both highlight crucial differences in the network's behavior if compared with the Ising SK model: as the temperature is lowered, no phase transitions are evidenced and the model continuously moves from a random (ergodic) behavior to a disordered (glassy) phase.