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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
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by Bernhard Krön and Elmar Teufl PDF
Trans. Amer. Math. Soc. 356 (2004), 393-414 Request permission

Abstract:

It is shown explicitly how self-similar graphs can be obtained as ‘blow-up’ constructions of finite cell graphs $\hat C$. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph $\hat C$, starting at a vertex $v$ of the boundary of $\hat C$. It is proved that the expected number of returns to $v$ before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the $n$-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the $n$-step transition probabilities.
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Additional Information
  • Bernhard Krön
  • Affiliation: Erwin Schrödinger Institute (ESI) Vienna, Boltzmanngasse 9, 1090 Wien, Austria
  • Address at time of publication: Department of Mathematics, University of Vienna, Strudlhofgasse 4, 1090 Wien, Austria
  • Email: bernhard.kroen@univie.ac.at
  • Elmar Teufl
  • Affiliation: Department of Mathematics C, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
  • Email: elmar.teufl@tugraz.at
  • Received by editor(s): August 19, 2002
  • Received by editor(s) in revised form: March 9, 2003
  • Published electronically: August 21, 2003
  • Additional Notes: The first author was supported by the project P14379-MAT of the Austrian Science Fund (FWF)
    The second author was supported by the START-project Y96-MAT of the FWF
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 393-414
  • MSC (2000): Primary 60J10; Secondary 05A15, 30D05
  • DOI: https://doi.org/10.1090/S0002-9947-03-03352-X
  • MathSciNet review: 2020038