We consider directed last passage percolation on \({\mathbb {Z}}^2\) with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture in [28], we show that for the passage times from the line \(x+y=0\) to the points (r, r) and (n, n), denoted \(X_{r}\) and \(X_{n}\) respectively, as \(n\rightarrow \infty \) and \(\frac{r}{n}\) is small but bounded away from zero, the covariance satisfies

thereby establishing \(\frac{4}{3}\) as the temporal covariance exponent. This differs from the corresponding exponent for the droplet initial condition recently rigorously established in [3, 27] and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.

我们考虑在\（{\ mathbb {Z}} ^ 2 \）上进行定向最后通过渗流，并在顶点上采用指数通过时间。随着端点在空间和时间上的变化，大地测量线的权重的耦合结构成为一个非常令人感兴趣的话题。一种特殊的专业是当人们将测地线视为从给定的初始数据开始沿时间方向变化的点时。本文考虑了平坦的初始条件，该条件对应于线对点的最后通过时间。在[28]中解决一个猜想，我们表明，对于从线\（x + y = 0 \）到点（r，  r）和（n，  n）的通过时间，表示为\（X_ {r} \ ）和\（X_ {n} \）分别由于\（n \ rightarrow \ infty \）和\（\ frac {r} {n} \）很小，但有界于零，所以协方差满足

从而将\（\ frac {4} {3} \）建立为时间协方差指数。这与最近在[3，27]中严格建立的液滴初始状态的相应指数不同，并且需要新颖的论据。关键成分包括对测地线几何的理解以及使用Brownian Gibbs属性将测地线重量轮廓与Brownian运动进行定量比较的最新进展。证明方法有望适用于更广泛的初始数据类别。