论文标题
最后一段渗透中的大地西瓜的交织和缩放指数
Interlacing and scaling exponents for the geodesic watermelon in last passage percolation
论文作者
论文摘要
在离散的平面最后一段渗透（LPP）中，随机值被独立分配给$ \ mathbb z^2 $中的每个顶点，并且在$ \ mathbb z^2 $中的每个有限立式路径归因于其角度值的总和。一系列不相交路径的重量是其成员重量的总和。大地测量的概念是两个顶点之间的最大重量路径，具有自然的概括性，涉及几个不相交的路径：$ k $ - 地形西瓜中的$ [1，n]^2 \ cap \ cap \ cap \ cap \ mathbb z^2 $是$ k $ discon的集合，在此广场中包含所有这些收藏品中最大的重量。尽管已知此类集合的权重是重要的对象，但最大化路径在很大程度上没有探索$ k = 1 $ case。对于诸如指数级和几何LPP之类的确切可解决的模型，众所周知，$ k = 1 $，管理重量波动和横向距离的指数为$ 1/3 $和$ 2/3 $；也就是说，通常，通过$ n^{1/3} $的订单，在$（n，n）$ $ n^{1/3} $的订单上，$（n，n）$在路线$（1,1）\ to（n，n）$上的重量;大地测量的最大欧几里得距离具有$ n^{2/3} $的顺序。假设对地球重量概况的强烈但局部形式的凸度和一个中等偏差的范围---在所有已知的可溶解型号中都可以使用---我们确定，通常，我们确定，通常，$ k $ - 地球上的西瓜的重量下降到$ k^$ k^{5/3} n^n^n^n^n^1/3} $ a n^$ k^$ k^$ k^$ by $μnk$下$ k^{1/3} n^{2/3} $。我们的论点至关重要的是依靠并发展了西瓜承认的非凡的确定性交错财产。我们的方法还对自然相关的点过程产生了急剧的刚度估计，从而改善了从确定点过程中可在可集成环境中可用的确定点过程中获得的估计。
In discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in $\mathbb Z^2$, and each finite upright path in $\mathbb Z^2$ is ascribed the weight given by the sum of values of its vertices. The weight of a collection of disjoint paths is the sum of its members' weights. The notion of a geodesic, a maximum weight path between two vertices, has a natural generalization concerning several disjoint paths: a $k$-geodesic watermelon in $[1,n]^2\cap\mathbb Z^2$ is a collection of $k$ disjoint paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have been largely unexplored beyond the $k=1$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $k=1$ the exponents that govern fluctuation in weight and transversal distance are $1/3$ and $2/3$; that is, typically, the weight of the geodesic on the route $(1,1) \to (n,n)$ fluctuates around a dominant linear growth of the form $μn$ by the order of $n^{1/3}$; and the maximum Euclidean distance of the geodesic from the diagonal has order $n^{2/3}$. Assuming a strong but local form of convexity and one-point moderate deviation bounds for the geodesic weight profile---which are available in all known exactly solvable models---we establish that, typically, the $k$-geodesic watermelon's weight falls below $μnk$ by order $k^{5/3}n^{1/3}$, and its transversal fluctuation is of order $k^{1/3}n^{2/3}$. Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes, which improve on estimates obtained via tools from the theory of determinantal point processes available in the integrable setting.
