### Abstract

On the complete graph K_{M} with M ≥ 3 vertices consider two independent discrete time random walks X and Y, choosing their steps uniformly at random. A pair of trajectories X = {X_{1}, X_{2}, ...} and Y = {Y_{1}, Y_{2}, ...} is called non-colliding, if by delaying their jump times one can keep both walks at distinct vertices forever. It was conjectured by P. Winkler that for large enough M the set of pairs of non-colliding trajectories {X, Y} has positive measure. N. Alon translated this problem to the language of coordinate percolation, a class of dependent percolation models, which in most situations is not tractable by methods of Bernoulli percolation. In this representation Winkler’s conjecture is equivalent to the existence of an infinite open cluster for large enough M. In this paper we establish the conjecture building upon the renormalization techniques developed in [4].

Original language | English (US) |
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Title of host publication | Sojourns in Probability Theory and Statistical Physics - III - Interacting Particle Systems and Random Walks, A Festschrift for Charles M. Newman |

Editors | Vladas Sidoravicius |

Publisher | Springer |

Pages | 90-137 |

Number of pages | 48 |

ISBN (Print) | 9789811503016 |

DOIs | |

State | Published - 2019 |

Event | International Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China Duration: Mar 25 2016 → Mar 27 2016 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 300 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | International Conference on Probability Theory and Statistical Physics, 2016 |
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Country | China |

City | Shanghai |

Period | 3/25/16 → 3/27/16 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Dependent percolation
- Renormalization

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## Cite this

*Sojourns in Probability Theory and Statistical Physics - III - Interacting Particle Systems and Random Walks, A Festschrift for Charles M. Newman*(pp. 90-137). (Springer Proceedings in Mathematics and Statistics; Vol. 300). Springer. https://doi.org/10.1007/978-981-15-0302-3_4