Random walks on periodic graphs

Kazami, Takahiro; Uchiyama, Kôhei

doi:10.1090/S0002-9947-08-04451-6

Random walks on periodic graphs
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by Takahiro Kazami and Kôhei Uchiyama PDF

Trans. Amer. Math. Soc. 360 (2008), 6065-6087 Request permission

Abstract:

This paper concerns random walks on periodic graphs embedded in the $d$-dimensional Euclidian space $\mathbf {R}^d$ and obtains asymptotic expansions of the Green functions of them up to the second order term, which, expressed fairly explicitly, are easily computable for many examples. The result is used to derive an asymptotic form of the hitting distribution of a hyperplane of co-dimension one, which involves not only the first but also second order terms of the expansion of the Green function. We also give similar expansions of the transition probabilities of the walks.

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