Coarsening with a frozen vertex

TL;DR

This paper investigates a modified coarsening model on a 2D lattice where one vertex is permanently frozen, demonstrating that all other vertices still flip infinitely often despite this constraint.

Contribution

It introduces and analyzes a new variant of the coarsening model with a frozen vertex, showing that infinite flipping persists for all other sites.

Findings

All other vertices flip infinitely often despite the frozen vertex.

The proof uses stochastic domination and influence propagation techniques.

The model extends understanding of coarsening dynamics with fixed boundary conditions.

Abstract

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z}^2}$ and initial state chosen from symmetric product measure, it is known (see~\cite{NNS}) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.