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

2018/10/16 by Kohei Yamamoto, Yamamoto, Kohei
Computer Science · Mathematics · #Advanced Graph Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1810.07162

openalex publication_date 2018/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 α(p). The function α(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 α(p). In other words, we construct another definition of the critical probability.

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