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Critical probability on the product graph of a regular tree and a line. (arXiv:1810.07162v1 [math.PR])

来源于:arXiv
We consider Bernoulli bond percolation on the product graph of a regular tree and a line. Schonmann showed that there are a.s. infinitely many infinite clusters at $p=p_u$ by using a certain function $\alpha(p)$. The function $\alpha(p)$ is defined by a exponential decay rate of probability that two vertices of the same layer are connected. We show the critical probability $p_c$ can be written by using $\alpha(p)$. In other words, we construct another definition of the critical probability. 查看全文>>