在$\mathbb{Z}^d$和$\mathbb{T}_d$上的空间宿主-寄生虫模型的相变
A phase transition for a spatial host-parasite model with extreme host immunities on $\mathbb{Z}^d$ and $\mathbb{T}_d$
AI总结：
研究了在$\mathbb{Z}^d$和$\mathbb{T}_d$上的空间宿主-寄生虫模型，揭示了寄生虫生存概率的相变现象以及图结构对临界值的影响。
摘要：
我们研究了一个寄生虫种群侵袭在图上分布的静止宿主的模型，这是青蛙模型的一种修改。每个宿主以概率$1-p$具有不可破坏的免疫力，寄生虫以简单对称随机游走的方式移动，试图感染遇到的任何宿主并随后繁殖自己。我们证明，在$\mathbb{Z}^d$（$d\ge 2$）和$d$-正则树$\mathbb{T}_d$（$d\ge 3$）上，寄生虫的生存概率在临界值$p_c\in(0,1)$处表现出相变。此外，我们证明在一般情况下，向基础图中添加顶点和边可能同时增加或减少$p_c$的值。最后，我们证明在准顶点传递图上，在温和的寄生虫子代分布假设下，一个固定顶点被寄生虫以概率$1$只被访问有限次数。
英文摘要：
We investigate a model of a parasite population invading spatially distributed immobile hosts on a graph, which is a modification of the frog model. Each host has an unbreakable immunity against infection with a certain probability $1-p$ and parasites move as simple symmetric random walks attempting to infect any host they encounter and subsequently reproduce themselves. We show that, on $\mathbb{Z}^d$ with $d\ge 2$ and the $d$-regular tree $\mathbb{T}_d$ with $d\ge 3$, the survival probability of parasites exhibits a phase transition at a critical value of $p_c\in(0,1)$. Also, we show that adding vertices and edges to the underlying graph can, in general, both increase or decrease the value of $p_c$. Finally, we show that on quasi-vertex-transitive graphs, with probability $1$, a fixed vertex is only visited finitely often by a parasite under mild assumptions on the offspring distribution of parasites.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
