论文标题
固定或不固定在两种an灭分支的随机步行中
To fixate or not to fixate in two-type annihilating branching random walks
论文作者
论文摘要
我们研究了两种类型之间的竞争模型，它们会随着$ \ mathbb {z}^d $的分支随机步行而发展。这两种类型分别由红球和蓝色球表示，该规则是接触时不同颜色的球会消灭。我们考虑$ \ mathbb {z}^d $的站点每个都包含一个球，每个球都具有一个独立的红色，否则$ p $和蓝色。我们解决了\ emph {fixation}的问题，指的是最终定居给定颜色的站点。在分支规则的温和力矩条件下，我们证明该过程几乎肯定会以$ p \ neq 1/2 $固定，并且每个站点的颜色都会无限地改变颜色，几乎肯定会肯定对于平衡的初始条件$ p = 1/2 $。
We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the process will fixate almost surely for $p\neq 1/2$, and that every site will change colour infinitely often almost surely for the balanced initial condition $p=1/2$.
