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| An Elementary Proof for the Recurrence of the Product Graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\} |
| ZHU Lin, YAO Qiang |
| Institute of Statistics and Big Data, Renmin University of China, Beijing, 100872, China; School of Statistics, East China Normal University,Shanghai, 200241, China |
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Abstract It is well known that the two dimensional integer lattice \mathbb{Z}^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\} is recurrent. The proof approach only utilizes the elementary methods in probability theory (without the
words of electric networks).
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