Time Correlation Exponents in Last Passage Percolation

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.