论文标题
指数级通道的当地平稳性
Local stationarity of exponential last passage percolation
论文作者
论文摘要
我们将最后一个通道的最后一段时间考虑到大小$Δn^{\ frac {2} {3}} $的每个顶点，距离起点$ n $。证明该社区的最后一次通道时间的增量与其固定版本相同，其概率很高，仅取决于$δ$。在此结果的帮助下，我们表明 1）$ \ text {airy} _2 $进程在本地接近总变化的布朗运动； 2）从每个顶点开始，从每个顶点开始，一个侧面长度$Δn^{\ frac {\ frac {2} {3}} $在距离$ n $的点上与一个盒子内的一个点同意，而无限的地理位生的树在同一方向上也是如此； 3）两个大地测量学从$ n^{\ frac {2} {3}} $彼此之间，到距离$ n $处的一个点不会在宏观尺度上靠近任何一个端点。 我们的主要结果仅取决于概率方法。
We consider point to point last passage times to every vertex in a neighbourhood of size $δN^{\frac{2}{3}}$, 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 $δ$ 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 $δN^{\frac{2}{3}}$ 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.
