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Probability Theory and Related Fields

, Volume 82, Issue 3, pp 435–449 | Cite as

Random Walks on graphs, electric networks and fractals

  • A. Telcs
Article

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 X0=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 dΩ is the exponent of the growth of the resistance of B N . Though the method of the paper was developed for the case when X0=x is a fixed vertex of the graph, we hope that the result can be generalized.

Keywords

Stochastic Process Fractal Dimension Random Walk Probability Theory Statistical Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • A. Telcs
    • 1
  1. 1.Library of Hungarian Academy of SciencesBudapestHungary

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