$\u2124^d$中的团-团模型
Cluster-Cluster model in $\mathbb{Z}^d$
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中文总结 AI 辅助
该研究针对$\mathbb{Z}^d$上的团-团随机游走合并过程，刻画了不同$\alpha$取值下有限时间是否形成无穷团的相变行为，给出一维精确相图。
中文摘要 AI 辅助
我们研究维数$d\geq1$时定义在$\mathbb{Z}^d$上的一类随机过程。给定平移不变且遍历的有限团初始构型，每个团$C$以速率$|C|^{-\alpha}$进行连续时间简单随机游走。若某个团试图移动到被另一个团占据的顶点，它不会移动，而是通过一条新边与对方团连接。在所有维数下我们证明：若$\alpha\ge0$，几乎必然不会在有限时间内自发形成无穷团；若$\alpha\le-1-2/d$，则几乎必然发生有限时间爆胀。在$\alpha\in(-1,0)$的区间内，系统行为高度依赖初始构型。此外，我们得到了一维情形下的精确相图。
英文摘要
We consider a stochastic process on $\mathbb{Z}^d$ for $d \geq 1$. Given a translation invariant and ergodic starting configuration of finite clusters, each cluster $C$ performs a continuous time simple random walk with rate $|C|^{-α}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. In all dimensions, we show that if $α\ge 0$, there is almost surely no spontaneous creation of an infinite cluster within finite time. Moreover, for any $α\le-1-2/d$ there is a finite-time blowup almost surely. In the regime $α\in(-1,0)$ we show that the behavior greatly depends on the initial configuration. In addition, in dimension one, we establish the exact phase diagram.
