Mean field dilute ferromagnet I. High temperature and zero temperature behavior

TL;DR

This paper analyzes the mean field dilute ferromagnet model, deriving free energy expressions at high and zero temperatures, identifying the critical line, and exploring zero-temperature entropy and phase behavior.

Contribution

It provides exact formulas for free energy and entropy, characterizes the phase transition, and offers a complete solution for the annealed model in both Poisson and Bernoulli cases.

Findings

Derived free energy expressions at high and zero temperatures.

Identified the critical line separating phases.

Computed positive entropy at zero temperature.

Abstract

We study the mean field dilute model of a ferromagnet. We find and prove an expression for the free energy density at high temperature, and at temperature zero. We find the critical line of the model, separating the phase with zero magnetization from the phase with symmetry breaking. We also compute exactly the entropy at temperature zero, which is strictly positive. The physical behavior at temperature zero is very interesting and related to infinite dimensional percolation, and suggests possible behaviors at generic low temperatures. Lastly, we provide a complete solution for the annealed model. Our results hold both for the Poisson and the Bernoulli versions of the model.