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In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusNon-Optimality of Invaded Geodesics in 2d Critical First-Passage Percolation

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Non-Optimality of Invaded Geodesics... [We study the critical case of first-passage percolation in two dimensions. Letting (te) be i.i.d. nonnegative weights assigned to the edges of 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 ℙ(te=0)=1∕2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {P}(t_e=0)=1/2$$ \end{document}, consider the induced pseudometric (passage time) T(x, y) for vertices x, y. It was shown in [4] that the growth of the sequence 𝔼T(0,∂B(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E}T(0,\partial B(n))$$ \end{document} (where B(n) = [−n, n]2) has the same order (up to a constant factor) as the sequence 𝔼Tinv(0,∂B(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E}T^{\text{inv}}(0,\partial B(n))$$ \end{document}. This second passage time is the minimal total weight of any path from 0 to ∂B(n) that resides in a certain embedded invasion percolation cluster. In this paper, we show that this constant factor cannot be taken to be 1. That is, there exists c > 0 such that for all n, 𝔼Tinv(0,∂B(n))≥(1+c)𝔼T(0,∂B(n)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \mathbb {E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb {E}T(0,\partial B(n)). $$ \end{document} This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structure.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusNon-Optimality of Invaded Geodesics in 2d Critical 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.

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References (16)

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
299 –317
DOI
10.1007/978-3-030-60754-8_14
Publisher site
See Chapter on Publisher Site

Abstract

[We study the critical case of first-passage percolation in two dimensions. Letting (te) be i.i.d. nonnegative weights assigned to the edges of 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 ℙ(te=0)=1∕2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {P}(t_e=0)=1/2$$ \end{document}, consider the induced pseudometric (passage time) T(x, y) for vertices x, y. It was shown in [4] that the growth of the sequence 𝔼T(0,∂B(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E}T(0,\partial B(n))$$ \end{document} (where B(n) = [−n, n]2) has the same order (up to a constant factor) as the sequence 𝔼Tinv(0,∂B(n))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb {E}T^{\text{inv}}(0,\partial B(n))$$ \end{document}. This second passage time is the minimal total weight of any path from 0 to ∂B(n) that resides in a certain embedded invasion percolation cluster. In this paper, we show that this constant factor cannot be taken to be 1. That is, there exists c > 0 such that for all n, 𝔼Tinv(0,∂B(n))≥(1+c)𝔼T(0,∂B(n)).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \mathbb {E}T^{\text{inv}}(0,\partial B(n)) \geq (1+c) \mathbb {E}T(0,\partial B(n)). $$ \end{document} This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structure.]

Published: Nov 4, 2020

Keywords: First-passage percolation; Invasion percolation; Near-critical percolation

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