  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 by Ferrari and Spohn
(SIGMA, 2016), 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 $n\to
\infty$ and $\frac{r}{n}$ is small but bounded away from zero, the covariance
satisfies $$\mbox{Cov}(X_{r},X_{n})=\Theta\left((\frac{r}{n})^{4/3+o(1)}
n^{2/3}\right),$$ 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 Ferrari and Occelli (2018), Basu
and Ganguly (2018), 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.
