• DOI:10.1007/s00440-022-01178-1
• Corpus ID: 237353332

Transitions for exceptional times in dynamical first-passage percolation

@article{Damron2021TransitionsFE,
  title={Transitions for exceptional times in dynamical first-passage percolation},
  author={Michael Damron and Jack Hanson and David Harper and Wai-Kit Lam},
  journal={Probability Theory and Related Fields},
  year={2021},
  volume={185},
  pages={1039 - 1085}
}

• M. Damron, J. Hanson, +1 author Wai-Kit Lam
• Published 30 August 2021
• Mathematics
• Probability Theory and Related Fields

In first-passage percolation (FPP), we let (τv)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau _v)$$\end{document} be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of τv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _v$$\end{document}, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if ∑F-1(1/2+1/2k)=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum F^{-1}(1/2+1/2^k) = \infty $$\end{document}, then a.s. there are exceptional times at which the weight grows atypically, but if ∑k7/8F-1(1/2+1/2k)<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty $$\end{document}, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.