Fixation for Two-Dimensional $${\mathcal {U}}$$-Ising and $${\mathcal {U}}$$-Voter Dynamics

Given a finite family $\mathcal U$ of subsets $\mathbb Z^d\setminus \{0\}$, the U$-$voter\ dynamics$ in space configurations $\{+,-\}^{\mathbb Z^d}$ is defined as follows: every $v\in\mathbb Z^d$ has an independent exponential random clock, and when clock at $v$ rings, vertex chooses $X\in\mathcal uniformly random. If set $v+X$ entirely state $+$ (resp. $-$), then updates to otherwise nothing happens. The $critical\ probability$ $p_c^{\text{vot}}(\mathbb Z^d,\mathcal U)$ for this model infimum over $p$ such that system almost surely fixates initial states vertices are chosen independently be with probability $-$ $1-p$. We prove U)<1$ wide class families U$. moreover consider U$-Ising dynamics show also exhibits same phase transition.