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,…,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.

我们考虑\(\mathbb{Z}^{2}\)上具有独立且同分布权重的第一通道渗透，其共同分布在有限指数矩下绝对连续。假设极限形状具有超过 32 个极值点，我们证明起点和终点附近的测地线具有显着的重叠，除了端点附近的小部分外，所有测地线都会合并。该陈述是量化的，所涉及的量与测地线长度呈幂律依赖性。

结果导致本杰明-卡莱-施拉姆中点问题的定量解决。结果表明，两个给定点之间的测地线穿过给定边的概率小于点与边之间的距离的幂。

我们进一步证明了特定分布族满足极限形状假设。

最后，与 1965 年 Hammersley-Welsh 高速公路和小道问题相关，我们证明了从原点开始被无限测地线覆盖的平方 {− n ,…, n } ^{2的预期分数至多是}n的倒数次幂。该结果是在没有明确的极限形状假设的情况下获得的。