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Tree-indexed random walks on groups and first passage percolation
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  • Published: March 1994

Tree-indexed random walks on groups and first passage percolation

  • Itai Benjamini1 &
  • Yuval Peres2 nAff3 

Probability Theory and Related Fields volume 98, pages 91–112 (1994)Cite this article

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  • 48 Citations

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Summary

Suppose that i.i.d. random variables are attached to the edges of an infinite tree. When the tree is large enough, the partial sumsS σ along some of its infinite paths will exhibit behavior atypical for an ordinary random walk. This principle has appeared in works on branching random walks, first-passage percolation, and RWRE on trees. We establish further quantitative versions of this principle, which are applicable in these settings. In particular, different notions of speed for such a tree-indexed walk correspond to different dimension notions for trees. Finally, if the labeling variables take values in a group, then properties of the group (e.g., polynomial growth or a nontrivial Poisson boundary) are reflected in the sample-path behavior of the resulting tree-indexed walk.

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Author information

Author notes
  1. Yuval Peres

    Present address: Department of Statistics, University of California, 94720, Berkeley, CA, USA

Authors and Affiliations

  1. Mathematics Institute, The Hebrew University, Jerusalem

    Itai Benjamini

  2. Mathematics Department, Yale University, New Haven, Conn, USA

    Yuval Peres

Authors
  1. Itai Benjamini
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  2. Yuval Peres
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Additional information

Partially supported by a grant from the Landau Center for Mathematical Analysis

Partially supported by NSF grant DMS-921 3595

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Benjamini, I., Peres, Y. Tree-indexed random walks on groups and first passage percolation. Probab. Th. Rel. Fields 98, 91–112 (1994). https://doi.org/10.1007/BF01311350

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  • Received: 21 August 1992

  • Revised: 29 September 1993

  • Issue Date: March 1994

  • DOI: https://doi.org/10.1007/BF01311350

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Mathematics Subject Classification

  • 60J15
  • 60G50
  • 60G60
  • 60B15
  • 60K35
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