Local stationarity of exponential last passage percolation

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 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 $\text{Airy}_2$ process is locally close to a Brownian motion in total variation; 2) 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; 3) 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. Our main results rely on probabilistic methods only.