Lower Bound for Large Local Transversal Fluctuations of Geodesics in Last Passage Percolation

For exactly solvable models of planar last passage percolation, it is known that geodesics of length n exhibit transversal fluctuations at scale n2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n<^>{2/3}$$\end{document} and matching (up to exponents) upper and lower bounds for the tail probabilities are available. The local transversal fluctuations near the endpoints are expected to be much smaller; it is known that the transversal fluctuation up to distance r << n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \ll n$$\end{document} is typically of the order r2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r<^>{2/3}$$\end{document} and the probability that the fluctuation is larger than tr2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$tr<^>{2/3}$$\end{document} is at most Ce-ct3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ce<^>{-c{t}<^>3}$$\end{document}. In this note, we provide a short argument establishing a matching lower bound for this probability.