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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusGeodesic Rays and Exponents in Ergodic Planar First Passage Percolation

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Geodesic Rays and Exponents in... [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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusGeodesic Rays and Exponents in Ergodic Planar First Passage Percolation

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.
Brito, Gerandy; Hoffman, Christopher
In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius — Nov 4, 2020
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References (10)

Publisher
Springer International Publishing
Copyright
© 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
163 –186
DOI
10.1007/978-3-030-60754-8_8
Publisher site
See Chapter on Publisher Site

Abstract

[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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