Probability Theory and Related Fields

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

Random Walks on graphs, electric networks and fractals

  • A. Telcs


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


Stochastic Process Fractal Dimension Random Walk Probability Theory Statistical Theory 
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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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