## Abstract

If W is the simple random walk on the square lattice Z^{2}, then W induces a random walk W_{G} on any spanning subgraph G ⊂ Z^{2} of the lattice as follows: viewing W as a uniformly random infinite word on the alphabet {x, −x, y, −y}, the walk W_{G} starts at the origin and follows the directions specified by W, only accepting steps of W along which the walk W_{G} does not exit G. For any fixed G ⊂ Z^{2}, the walk W_{G} is distributed as the simple random walk on G, and hence W_{G} is almost surely recurrent in the sense that W_{G} visits every site reachable from the origin in G infinitely often. This fact naturally leads us to ask the following: does W almost surely have the property that W_{G} is recurrent for every G ⊂ Z^{2}? We answer this question negatively, demonstrating that exceptional subgraphs exist almost surely. In fact, we show more to be true: exceptional subgraphs continue to exist almost surely for a countable collection of independent simple random walks, but on the other hand, there are almost surely no exceptional subgraphs for a branching random walk.

Original language | English (US) |
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Pages (from-to) | 2017-2027 |

Number of pages | 11 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2020 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Exceptional graphs
- Random walks
- Recurrence and transience
- Traversal sequences