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Random Walks on graphs, electric networks and fractals
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  • Published: August 1989

Random Walks on graphs, electric networks and fractals

  • A. Telcs1 

Probability Theory and Related Fields volume 82, pages 435–449 (1989)Cite this article

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Summary

This paper is devoted to the problem of the nearest, neighbour Random Walk on infinite graphs. We investigate the RW X n started from a fixed vertex X 0=x∈V of the graph G=(V, E) and the expected value of the first exit time T N from the N-ball B N in G. It will be shown that if G is sufficiently “regular” then

$$d\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } d_\Omega ,$$
$$d_R \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2$$

and

$$d_R = d + 2 - d_\Omega $$

where d R is the RW dimension, d is the fractal dimension and d Ω is the exponent of the growth of the resistance of B N . Though the method of the paper was developed for the case when X 0=x is a fixed vertex of the graph, we hope that the result can be generalized.

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Authors and Affiliations

  1. Library of Hungarian Academy of Sciences, P.O. Box 7, 1361, Budapest, Hungary

    A. Telcs

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  1. A. Telcs
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Telcs, A. Random Walks on graphs, electric networks and fractals. Probab. Th. Rel. Fields 82, 435–449 (1989). https://doi.org/10.1007/BF00339997

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  • Received: 15 February 1988

  • Issue Date: August 1989

  • DOI: https://doi.org/10.1007/BF00339997

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Keywords

  • Stochastic Process
  • Fractal Dimension
  • Random Walk
  • Probability Theory
  • Statistical Theory
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