Recurrence of multiply-ended planar triangulations

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In this note we show that a bounded degree planar triangulation is recurrent if and only if the set of accumulation points of some/any circle packing of it is polar (that is, planar Brownian motion avoids it with probability 1). This generalizes a theorem of He and Schramm [6] who proved it when the set of accumulation points is either empty or a Jordan curve, in which case the graph has one end. We also show that this statement holds for any straight-line embedding with angles uniformly bounded away from 0.

Original languageEnglish
Article numberA05
Pages (from-to)1-6
Number of pages6
JournalElectronic Communications in Probability
StatePublished - 2017


  • Circle packing
  • Random walk

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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