朱琳, 姚强. 乘积图\mathbb{Z}^2\times\{0,1,\ldots,l-1\}的常返性的初等证明[J]. 应用概率统计, 2018, 34(3): 275-283. DOI: 10.3969/j.issn.1001-4268.2018.03.005
引用本文: 朱琳, 姚强. 乘积图\mathbb{Z}^2\times\{0,1,\ldots,l-1\}的常返性的初等证明[J]. 应用概率统计, 2018, 34(3): 275-283. DOI: 10.3969/j.issn.1001-4268.2018.03.005
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ZHU Lin, YAO Qiang. An Elementary Proof for the Recurrence of the Product Graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\}[J]. Chinese Journal of Applied Probability and Statistics, 2018, 34(3): 275-283. DOI: 10.3969/j.issn.1001-4268.2018.03.005
Citation: ZHU Lin, YAO Qiang. An Elementary Proof for the Recurrence of the Product Graph \mathbb{Z}^2\times\{0,1,\ldots,l-1\}[J]. Chinese Journal of Applied Probability and Statistics, 2018, 34(3): 275-283. DOI: 10.3969/j.issn.1001-4268.2018.03.005
shu

乘积图\mathbbZ^2\times\0,1,\ldots,l-1\的常返性的初等证明

An Elementary Proof for the Recurrence of the Product Graph \mathbbZ^2\times\0,1,\ldots,l-1\

    摘要
  • 摘要: 我们已知二维整数格点\mathbbZ^2是常返的,而三维整数格点\mathbbZ^3是非常返的. 本文严格证明了二维整数格点\mathbbZ^2与有限线段\0,1,\ldots,l-1\的乘积图是常返的.证明过程只用到了概率论中的初等方法而没有用到电网络的术语.

     

    Abstract: It is well known that the two dimensional integer lattice \mathbbZ^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbbZ^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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