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

Given a finite family $\mathcal{U}$ of finite subsets of $\mathbb Z^d\setminus {0}$,the$\mathcal U$-$voter\ dynamics$inthespaceofconfigurations${+,-}^{\mathbb Z^d}$isdefinedasfollows: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 $criticalprobability$ $p_c^{\text{vot}}({\mathbb{Z}}^d,\mathcal{U})$ for this model is the infimum over $p$ such that this system almost surely fixates at $+$ when the initial states for the vertices are chosen independently to be $+$ with probability $p$ and to be $-$ with probability $1-p$. We prove that $p_c^{$\text{vot}$}(\mathbb Z^d,\mathcal U)<1$forawideclassoffamilies$\mathcal U$. We moreover consider the $\mathcal{U}$-Ising dynamics and show that this model also exhibits the same phase transition.