Critical probability on the product graph of a regular tree and a line

TL;DR

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Contribution

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Abstract

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.