We consider point-to-point last-passage times to every vertex in a neighbourhood of size \(\delta N^{\nicefrac {2}{3}}\) 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 \(\text {Airy}_2\) 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 \(\delta N^{\nicefrac {2}{3}}\) 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 \(N^{\nicefrac {2}{3}}\) 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.

我们考虑到距起始点距离N处大小为\（\ delta N ^ {\ nicefrac {2} {3}} \）的邻域中每个顶点的点对点最后通过时间。在该邻域中，最后通过时间的增量被证明共同等于其固定版本，并且仅取决于\（\ delta \）的可能性很高。通过此结果，我们表明（1）\（\ text {Airy} _2 \）过程在局部变化上局部接近布朗运动；（2）边长为\（\ delta N ^ {\ nicefrac {2} {3}} \）的盒子中的每个顶点到距离N的点的点对点测地线树在框内与沿相同方向行进的半无限大地测量树相吻合；（3）2点至点在距离开始测地线\（N ^ {\ nicefrac {2} {3}} \）彼此，以在距离的点Ñ，不会聚结靠近任一端点上的刻度ñ。我们的主要结果仅依赖于概率方法。