Equilibrium Biphasicity and Non-Binary Pathwise Confinement in Stochastic Ising Models

TL;DR

This paper explores the complex structure of the Glauber dynamics in the low-temperature 2D Ising model, revealing a dense set of confined trajectories that do not form a new Gibbs phase but enrich the understanding of the system's equilibrium states.

Contribution

It demonstrates the existence of a dense family of initial configurations with confined trajectories, showing the pathwise structure exceeds the Gibbs-phase classification.

Findings

Confined trajectories are dense in the state space.

Cesaro averages of these trajectories converge to a mixture of pure phases.

The pathwise structure is richer than the Gibbs phases without introducing a new phase.

Abstract

For the low-temperature two-dimensional Ising model, the two pure Gibbs phases exhaust the extremal equilibrium states, but not the pathwise absorbing structure of the Glauber dynamics. Let \[ P^\pm=\{\sigma:M_n(\sigma)\to \pm m_\beta\},\qquad R=\Omega\setminus(P^+\cup P^-). \] We show that \(R\) is null under both pure phases but contains a dense pathwise confined subset. More precisely, we construct a dense family of initial configurations whose trajectories are confined to the centered sector \[ C_0=\{\sigma:M_n(\sigma)\to0\}\subset R. \] Nevertheless, the corresponding Cesaro averages converge to \(\frac12(\mu^++\mu^-)\). Thus the pathwise absorbing geometry is richer than the Gibbs-phase classification, without creating a third Gibbs phase.