A hierarchical mean field model of interacting spins

TL;DR

This paper develops a hierarchical mean field model combining interacting spins and diffusions, analyzing phase transitions and macroscopic limits across different regimes and extending to multiple hierarchical levels.

Contribution

It introduces a novel hierarchical mean field framework integrating spin-flip dynamics with diffusive variables, extending analysis to multiple levels and regimes.

Findings

Macroscopic limits identified at various scales

Phase transitions characterized in subcritical regimes

Heuristic analysis of supercritical zero-temperature behavior

Abstract

We consider a system of hierarchical interacting spins under dynamics of spin-flip type with a ferromagnetic mean field interaction, scaling with the hierarchical distance, coupled with a system of linearly interacting hierarchical diffusions of Ornstein-Uhlenbeck type. In particular, the diffusive variables enter in the spin-flip rates, effectively acting as dynamical magnetic fields. In absence of the diffusions, the spin-flip dynamics can be thought of as a modification of the Curie--Weiss model. We study the mean field and the two-level hierarchical model, in the latter case restricting to a subcritical regime, corresponding to high temperatures, obtaining macroscopic limits at different spatio-temporal scales and studying the phase transitions in the system. We also formulate a generalization of our results to the $k$-th level hierarchical case, for any $k$ finite, in the subcritical regime. We finally address the supercritical regime, in the zero-temperature limit, for the two-level hierarchical case, proceeding heuristically with the support of numerics.