论文标题
平面第一-Pasgage Percolation中的测量学和BKS中点问题的合并
Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation
论文作者
论文摘要
我们考虑$ \ Mathbb z^2 $具有独立且分布相同的权重的第一本处理渗透，其共同分布绝对是连续的，并且具有有限的指数力矩。假设极限形状具有超过32个极端点，我们证明，在附近的起点和终点的地理学具有显着的重叠，并且在其终点附近的小部分外也有融合。该声明被量化，涉及数量对大地测量长度的幂律依赖性。 结果导致了Benjamini-Kalai-Schramm中点问题的定量分辨率。结果表明，两个给定点之间的大地测量通过给定边缘的概率小于点和边缘之间的距离的功率。 我们进一步证明，对于特定的分布家族，满足了极限形状假设。 最后，与1965年的Hammersley（Welsh Highways and Byways问题）有关，我们证明了广场$ \ { - n，\ dots，n \}^2 $的预期部分，该$ \}^2 $由从原点开始的无限大地测量学覆盖，最多是$ n $的倒数功率。在没有明确的极限形状假设的情况下获得此结果。
We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.
