Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices

TL;DR

This paper investigates how frozen vertices influence the long-term behavior of a majority-vote Markov process on a lattice, revealing conditions under which all sites fixate or certain states do not percolate.

Contribution

It provides new insights into the effects of randomly placed frozen vertices on the fixation and percolation properties of majority-vote dynamics on bZ^d.

Findings

All sites fixate to plus when b ho^+ > 0 and b ho^- = 0.

Fixed minus and flippers do not percolate when b ho^+ > 0 and b ho^- is very small.

Results extend to certain deterministic arrangements of frozen vertices.

Abstract

We study Markov processes in which $\pm 1$-valued random variables $\sigma_x(t), x\in \mathbb{Z}^d$, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $\rho^+$ (resp., $\rho^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $\rho^+ >0$ and $\rho^- =0$, all sites are fixed plus, while for $\rho^+ >0$ and $\rho^-$ very small (compared to $\rho^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.