  For directed last passage percolation on $\mathbb{Z}^2$ with exponential
passage times on the vertices, let $T_{n}$ denote the last passage time from
$(0,0)$ to $(n,n)$. We consider asymptotic two point correlation functions of
the sequence $T_{n}$. In particular we consider ${\rm Corr}(T_{n}, T_{r})$ for
$r\le n$ where $r,n\to \infty$ with $r\ll n$ or $n-r \ll n$. We show that in
the former case ${\rm Corr}(T_{n}, T_{r})=\Theta((\frac{r}{n})^{1/3})$ whereas
in the latter case $1-{\rm Corr}(T_{n}, T_{r})=\Theta ((\frac{n-r}{n})^{2/3})$.
The argument revolves around finer understanding of polymer geometry and is
expected to go through for a larger class of integrable models of last passage
percolation. As by-products of the proof, we also get a couple of other results
of independent interest: Quantitative estimates for locally Brownian nature of
pre-limits of Airy$_2$ process coming from exponential LPP, and precise
variance estimates for lengths of polymers constrained to be inside thin
rectangles at the transversal fluctuation scale.
