A hierarchical mean field model of interacting spins

Paolo Dai Pra, Marco Formentin and Guglielmo Pelino

Stochastic Processes and their Applications, 2021, vol. 140, issue C, 287-338

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 kth 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.

Keywords: Mean field interacting particle systems; Hierarchical spin systems; Curie–weiss model; Propagation of chaos (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.spa.2021.06.011

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