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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Random walks on periodic graphs


Authors: Takahiro Kazami and Kôhei Uchiyama
Journal: Trans. Amer. Math. Soc. 360 (2008), 6065-6087
MSC (2000): Primary 60G50; Secondary 60J45
Published electronically: June 16, 2008
MathSciNet review: 2425703
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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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Additional Information

Takahiro Kazami
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro Tokyo, 152-8551 Japan
Email: uchiyama@math.titech.ac.jp

Kôhei Uchiyama
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro Tokyo, 152-8551 Japan

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04451-6
PII: S 0002-9947(08)04451-6
Keywords: Asymptotic expansion, Markov additive process, periodic graph, Green function, hitting distribution of a line
Received by editor(s): July 26, 2006
Received by editor(s) in revised form: November 21, 2006
Published electronically: June 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.