Random and automatic walks

  • F. M. Dekking
Conference paper
Part of the Centre de Physique des Houches book series (LHWINTER, volume 3)

Abstract

The symmetric simple random walk in ℤ d describes the movement of a particle on the d-dimensional lattice which, independently of its past, moves at time instants 1, 2,... with equal probability to one of its 2d neighbours. Here, the particle is at the origin at time O. The fundamental question about the behaviour of the particle is whether its walk is recurrent: i.e., it returns to the origin infinitely often (with probability one), or transient, i.e., eventually the walk will leave any ball around the origin (with probability one). The answer has already been given by Polya in 1921: if d ≤ 2 then the walk is recurrent, if d ≥ 3 then it is transient [25].

Keywords

Covariance 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • F. M. Dekking
    • 1
  1. 1.Department of Statistics, Probability and Operations ResearchTU Delft, Faculty of Mathematics and InformaticsDelftThe Netherlands

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