论文标题
损坏的指南针模型的相位过渡的清晰度
Sharpness of the Phase Transition for the Corrupted Compass Model on Transitive Graphs
论文作者
论文摘要
在顶点传递图上的损坏的指南针模型中，每个顶点的相邻边缘是随机选择并打开的。此外，使用概率$ p $，每个顶点独立，每个相邻的边缘都会打开。我们研究此模型中开放簇的大小。 Hirsch等。已经表明，对于小$ p $，所有开放式群集几乎都是有限的，而对于大$ p $，根据基础图，几乎肯定存在一个无限的开放式群集。我们表明，相应的相变很锐利，即在亚临界方面，所有开放簇呈指数型。此外，我们证明了超临界状态中的平均场下限。该证明使用现在使用OSS不等式建立的良好方法。该注释的第二个目标是在简单的设置中展示此方法。
In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability $p$, independently for every vertex, every neighbouring edge is opened. We study the size of open clusters in this model. Hirsch et al. have shown that for small $p$ all open clusters are finite almost surely, while for large $p$, depending on the underlying graph, there exists an infinite open cluster almost surely. We show that the corresponding phase transition is sharp, i.e., in the subcritical regime, all open clusters are exponentially small. Furthermore we prove a mean-field lower bound in the supercritical regime. The proof uses the by now well established method using the OSSS inequality. A second goal of this note is to showcase this method in an uncomplicated setting.
