Time Correlation Exponents in Last Passage Percolation

Abstract

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 Corr(T _n, T _r) for r ≤ n where r, n →∞ with r ≪ n or n − r ≪ n. Establishing a conjecture from Ferrari and Spohn (SIGMA 12:074, 2016), we show that in the former case $\mathrm {Corr}(T_{n}, T_{r})=\varTheta ((\frac {r}{n})^{1/3})$ whereas in the latter case $1-{\mathrm {Corr}}(T_{n}, T_{r})=\varTheta ((\frac {n-r}{n})^{2/3})$. 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 a by-product of the proof, we also get quantitative estimates for locally Brownian nature of pre-limits of Airy₂ process coming from exponential LPP, a result of independent interest.

In memory of Vladas Sidoravicius