A Phase Transition in a Quenched Amorphous Ferromagnet

TL;DR

This paper investigates the existence and multiplicity of quenched thermodynamic states in a randomly distributed amorphous ferromagnet, revealing phase transition phenomena influenced by particle density and temperature.

Contribution

It establishes the existence of multiple quenched thermodynamic states and their dependence on the underlying random particle configuration in an amorphous ferromagnet model.

Findings

Quenched thermodynamic states exist with probability one.

Multiple thermodynamic states occur at high density and temperature.

States depend measurably on the underlying point process realization.

Abstract

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over $\mathbb{R}^d$, $d\geq 2$. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the particle density (i.e., the intensity of the underlying point process) and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.