正则树上的接触过程渗流
Percolation of the contact process on the regular tree
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中文总结 AI 辅助
本文研究正则树上接触过程的渗流相变，证明在 $d\geq 3$ 时 $\lambda_1<\lambda_p$，并在 $d\geq 7$ 时给出完整的相变顺序，解决了长期未决问题。
中文摘要 AI 辅助
当 $d\geq 3$ 时，正则树 $\mathbb{T}_d$ 上的接触过程具有全局存活和局部存活两个相变，由 Pemantle 和 Liggett 在值 $\lambda_1$ 和 $\lambda_2$ 处发现。我们从每个顶点均被感染的状态出发，让系统松弛到所谓的上不变感染状态。在该平稳状态下，$\lambda_p$ 是感染顶点能够通过 $\mathbb{T}_d$ 渗流的临界值，而 $\lambda_{p^\complement}$ 是健康顶点能够渗流之前的参数。我们发现，当 $d\geq 7$ 时，在 $\mathbb{T}_d$ 上这两个都是不同的相变，满足 $0<\lambda_1<\lambda_p<\lambda_2<\lambda_{p^\complement}<+\infty$。这些比较中最有趣的是 $\lambda_1<\lambda_p$，我们对所有 $d\geq 3$ 都证明了该不等式。这一比较 $\lambda_1<\lambda_p$ 在 $\mathbb{Z}^d$（$d\geq 2$）上是一个长期未解决的公开问题，并且除 $\lambda_p$ 为无穷大的图之外，尚未在任何其他图上得到证明。
英文摘要
The contact process on the regular tree $\mathbb{T}_d$ when $d\geq 3$ has the two phase transitions of global and of local survival, found by Pemantle and Liggett at values $λ_1$ and $λ_2$. We start the system with every vertex infected and let it relax to what is known as the upper invariant infection. In this stationary state, $λ_p$ is the critical value beyond which the infected vertices can percolate through $\mathbb{T}_d$, and $λ_{p^\complement}$ is the parameter before which the healthy vertices can percolate. We find these are both distinct phase transitions \[0<λ_1<λ_p<λ_2<λ_{p^\complement}<+\infty\] on $\mathbb{T}_d$ when $d\geq 7$. The most interesting of these comparisons is $λ_1<λ_p$, which we find for all $d\geq 3$. This comparison $λ_1<λ_p$ is a long-standing open question on $\mathbb{Z}^d$ with $d\geq 2$ and was not yet found on any other graphs except where $λ_p$ is infinite.
发表机构
- CRiSM, Department of Statistics, University of Warwick（华威大学统计系CRiSM）
- Department of Mathematics, New York University Shanghai（纽约大学上海分校数学系）
机构由 AI 辅助整理，请以论文原文为准。
