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[We study first passage percolation on the plane for a family of invariant, ergodic measures 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}. We prove that for all of these models the asymptotic shape is the ℓ1 ball and that there are exactly four infinite geodesics starting at the origin a.s. In addition we determine the exponents for the variance and wandering of finite geodesics. We show that the variance and wandering exponents do not satisfy the relationship of χ = 2ξ − 1 which is expected for independent first passage percolation.]
Published: Nov 4, 2020
Keywords: First passage percolation; Fluctuation exponent; Variance exponent
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