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 Tn denote the last passage time from (0, 0) to (n, n). We consider asymptotic two point correlation functions of the sequence Tn. In particular we consider Corr(Tn, Tr) 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.