Thermodynamic Limit for Mean-Field Spin Models

TL;DR

This paper establishes a sufficient condition for the existence of the thermodynamic limit in mean-field spin models, including Curie-Weiss, p-spin, and Hopfield models, by demonstrating free energy density growth and subadditivity.

Contribution

It proves a general condition ensuring the thermodynamic limit for a broad class of mean-field spin models, extending previous results to models with randomness and finite patterns.

Findings

Condition verified for Curie-Weiss and p-spin models

Thermodynamic limit exists for all considered models

Free energy density increases with system size

Abstract

If the Boltzmann-Gibbs state $\omega_N$ of a mean-field $N$-particle system with Hamiltonian $H_N$ verifies the condition $$ \omega_N(H_N) \ge \omega_N(H_{N_1}+H_{N_2}) $$ for every decomposition $N_1+N_2=N$, then its free energy density increases with $N$. We prove such a condition for a wide class of spin models which includes the Curie-Weiss model, its p-spin generalizations (for both even and odd p), its random field version and also the finite pattern Hopfield model. For all these cases the existence of the thermodynamic limit by subadditivity and boundedness follows.