# In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusTime Correlation Exponents in Last Passage Percolation

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Time Correlation Exponents in Last... [For directed last passage percolation on ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {Z}^2$$ \end{document} 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 Corr(Tn,Tr)=Θ((rn)1∕3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathrm {Corr}(T_{n}, T_{r})=\varTheta ((\frac {r}{n})^{1/3})$$ \end{document} whereas in the latter case 1−Corr(Tn,Tr)=Θ((n−rn)2∕3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1-{\mathrm {Corr}}(T_{n}, T_{r})=\varTheta ((\frac {n-r}{n})^{2/3})$$ \end{document}. 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusTime Correlation Exponents in Last Passage Percolation

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.
22 pages

# References (27)

Publisher
Springer International Publishing
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
ISBN
978-3-030-60753-1
Pages
101 –123
DOI
10.1007/978-3-030-60754-8_5
Publisher site
See Chapter on Publisher Site

### Abstract

[For directed last passage percolation on ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {Z}^2$$ \end{document} 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 Corr(Tn,Tr)=Θ((rn)1∕3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathrm {Corr}(T_{n}, T_{r})=\varTheta ((\frac {r}{n})^{1/3})$$ \end{document} whereas in the latter case 1−Corr(Tn,Tr)=Θ((n−rn)2∕3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1-{\mathrm {Corr}}(T_{n}, T_{r})=\varTheta ((\frac {n-r}{n})^{2/3})$$ \end{document}. 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.]

Published: Nov 4, 2020

Keywords: Last passage percolation; Time correlation; KPZ universality class