AbstractWe consider point-to-point last-passage times to every vertex in a neighbourhood of size $$\delta N^{\nicefrac {2}{3}}$$
δ
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 $$\delta $$
δ
. Through this result we show that (1) the $$\text {Airy}_2$$
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 $$\delta N^{\nicefrac {2}{3}}$$
δ
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 $$N^{\nicefrac {2}{3}}$$
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.
A Law of the Iterated Logarithm for Directed Last Passage Percolation
