The components of the Wired Spanning Forest are recurrent

Abstract

 We show that a.s. all of the connected components of the Wired Spanning Forest are recurrent, proving a conjecture of Benjamini, Lyons, Peres and Schramm. Our analysis relies on a simple martingale involving the effective conductance between the endpoints of an edge in a uniform spanning tree. We believe that this martingale is of independent interest and will find further applications in the study of uniform spanning trees and forests.

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Received: 20 April 2001 / Revised version: 23 July 2002 / Published online: 14 November

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Morris, B. The components of the Wired Spanning Forest are recurrent. Probab Theory Relat Fields 125, 259–265 (2003). https://doi.org/10.1007/s00440-002-0236-0

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Keywords

  • Endpoint
  • Peris
  • Span Tree
  • Independent Interest
  • Effective Conductance