Recurrent Graphs where Two Independent Random Walks Collide Finitely Often

@article{Krishnapur2004RecurrentGW,
  title={Recurrent Graphs where Two Independent Random Walks Collide Finitely Often},
  author={Manjunath Krishnapur and Y. Peres},
  journal={Electronic Communications in Probability},
  year={2004},
  volume={9},
  pages={72-81}
}
  • Manjunath Krishnapur, Y. Peres
  • Published 2004
  • Mathematics
  • Electronic Communications in Probability
  • We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from $Z^2$ by removing all horizontal edges off the $x$-axis, has this property. We also conjecture that the same property holds for some other graphs, including the incipient infinite cluster for critical percolation in $Z^2$. 
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    Acknowledgement: We thank Jeffrey Steif and Nina Gantert for encouragement
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