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.

Authors and Affiliations

1. Statistics Department, University of California, Berkeley CA 94720, USA. Supported by an NSF Postdoctoral Fellowship. e-mail: morris@stat.berkeley.edu, , , , , , US

   Ben Morris

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

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