Local stationarity of exponential last passage percolation

Márton Balázs, Ofer Busani, Timo Seppäläinen

Published 2020-01-12Version 1

We consider point to point last passage times to every vertex in a neighbourhood of size $\delta N^{\frac{2}{3}}$, distance $N$ away from the starting point. The increments of these last passage times in this neighbourhood are shown to be \emph{jointly equal} to their stationary versions with high probability that depends on $\delta$ only. With the help of this result we show that 1) the tree of point to point geodesics starting from every vertex in a box of side length $\delta N^{\frac{2}{3}}$ going to a point at distance $N$ agree inside the box with the tree of infinite geodesics going in the same direction; 2) two geodesics starting from $N^{\frac{2}{3}}$ away from each other, to a point at distance $N$ will not coalesce too close to either endpoints on the macroscopic scale.