论文标题
二维$ \ Mathcal U $ $ $ $ - $ \ MATHCAL U $ -VOTER DYNAMITS的固定
Fixation for Two-Dimensional $\mathcal U$-ISING and $\mathcal U$-VOTER Dynamics
论文作者
论文摘要
给定有限的家庭$ \ MATHCAL U $的有限子集为$ \ Mathbb z^d \ setMinus \ {0 \} $，$ \ mathcal u $ - $ $ $ $选民\ dynamics $在配置空间中$ \ \ \ {+{+{+， - \}具有独立的指数随机时钟，当时钟处为$ v $戒指时，顶点$ v $选择$ x \ in \ mathcal u $均匀地随机。如果集合$ v+x $完全在状态$+$（分别$ - $）中，则$ v $更新到$+$（resp。$ - $），否则什么都不会发生。对于此型号而言，$ crialial \概率$ $ $ $ P_C^{\ text {fot}}（\ mathbb z^d，\ mathcal u）$是超过$ p $的aff th这个系统，以至于该系统几乎可以肯定地固定在$+$时，当对顶点的初始状态独立选择为$+$ $ p $ $ p $ $ $ $ $ $ -1-$ -1-P-1-p。我们证明$ p_c^{\ text {fot}}（\ mathbb z^d，\ mathcal u）<1 $ $ \ mathcal u $。此外，我们考虑了$ \ Mathcal U $的动力学，并表明该模型也表现出相同的相变。
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 $critical\ probability$ $p_c^{\text{vot}}(\mathbb Z^d,\mathcal U)$ 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 $p_c^{\text{vot}}(\mathbb Z^d,\mathcal U)<1$ for a wide class of families $\mathcal U$. We moreover consider the $\mathcal U$-Ising dynamics and show that this model also exhibits the same phase transition.
