## Abstract

We consider point-to-point last-passage times to every vertex in a neighbourhood of size delta*N^{2/3} at distance N from the starting point. The increments of the last-passage times in this neighbourhood are shown to be jointly equal to their stationary versions with high probability that depends only on delta. Through this result we show that

(1) the Airy_2 process is locally close to a Brownian motion in total variation;

(2) the tree of point-to-point geodesics from every vertex in a box of side length delta*N^{2/3} going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction;

(3) two point-to-point geodesics started at distance N^{2/3} from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N.

Our main results rely on probabilistic methods only.

(1) the Airy_2 process is locally close to a Brownian motion in total variation;

(2) the tree of point-to-point geodesics from every vertex in a box of side length delta*N^{2/3} going to a point at distance N agrees inside the box with the tree of semi-infinite geodesics going in the same direction;

(3) two point-to-point geodesics started at distance N^{2/3} from each other, to a point at distance N, will not coalesce close to either endpoint on the scale N.

Our main results rely on probabilistic methods only.

Original language | English |
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Pages (from-to) | 113-162 |

Number of pages | 50 |

Journal | Probability Theory and Related Fields |

Volume | 180 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 15 Mar 2021 |

### Bibliographical note

Funding Information:M. Balázs was partially supported by EPSRC’s EP/R021449/1 Standard Grant.

Funding Information:

T. Seppäläinen was partially supported by National Science Foundation grant DMS-1854619 and by the Wisconsin Alumni Research Foundation.

Publisher Copyright:

© 2021, The Author(s).