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Random mappings of scaled graphs
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  • Published: 08 May 2008

Random mappings of scaled graphs

  • Anna Erschler1 

Probability Theory and Related Fields volume 144, pages 543–579 (2009)Cite this article

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Abstract

Given a connected finite graph Γ with a fixed base point O and some graph G with a based point we study random 1-Lipschitz maps of a scaled Γ into G. We are mostly interested in the case where G is a Cayley graph of some finitely generated group, where the construction does not depend on the choice of base points. A particular case of Γ being a graph on two vertices and one edge corresponds to the random walk on G, and the case where Γ is a graph on two vertices and two edges joining them corresponds to Brownian bridge in G. We show, that unlike in the case \({G=\mathbb Z^d}\), the asymptotic behavior of a random scaled mapping of Γ into G may differ significantly from the asymptotic behavior of random walks or random loops in G. In particular, we show that this occurs when G is a free non-Abelian group. Also we consider the case when G is a wreath product of \({\mathbb Z}\) with a finite group. To treat this case we prove new estimates for transition probabilities in such wreath products. For any group G generated by a finite set S we define a functor E from category of finite connected graphs to the category of equivalence relations on such graphs. Given a finite connected graph Γ, the value E G,S (Γ) can be viewed as an asymptotic invariant of G.

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Bâtiment 425, 91405, Orsay Cedex, France

    Anna Erschler

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  1. Anna Erschler
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Correspondence to Anna Erschler.

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Erschler, A. Random mappings of scaled graphs. Probab. Theory Relat. Fields 144, 543–579 (2009). https://doi.org/10.1007/s00440-008-0154-x

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  • Received: 14 May 2007

  • Revised: 29 February 2008

  • Published: 08 May 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0154-x

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Keywords

  • Random walks on groups
  • Brownian bridges
  • Free groups
  • Lamplighter groups
  • Random graph homomorphism
  • Lipschitz maps

Mathematics Subject Classification (2000)

  • 20F69
  • 20F65
  • 20P05
  • 60B15
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