# Fixation for Two-Dimensional $\mathcal U$-ISING and $\mathcal U$-VOTER Dynamics

@article{Blanquicett2020FixationFT,
title={Fixation for Two-Dimensional \$\mathcal U\$-ISING and \$\mathcal U\$-VOTER Dynamics},
author={Daniel Blanquicett},
journal={arXiv: Probability},
year={2020}
}
Given a finite family $\mathcal U$ of finite subsets of $\mathbb Z^d\setminus \{0\}$, the $\mathcal U$-$voter\ dynamics$ in the space of configurations $\{+,-\}^{\mathbb Z^d}$ is defined as follows: every $v\in\mathbb Z^d$ has an independent exponential random clock, and when the clock at $v$ rings, the vertex $v$ chooses $X\in\mathcal U$ uniformly at random. If the set $v+X$ is entirely in state $+$ (resp. $-$), then the state of $v$ updates to $+$ (resp. $-$), otherwise nothing happens. The… Expand

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