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[The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in ℤd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb Z^d$$ \end{document}, depending on two integer parameters 1 ≤ d1, d2 ≤ d, which whenever it is at a site x∈ℤd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$x\in \mathbb Z^d$$ \end{document} at time n, it jumps to x ± ei with uniform probability, where e1, …, ed are the canonical vectors, for 1 ≤ i ≤ d1, if the site x was visited for the first time at time n, while it jumps to x ± ei with uniform probability, for 1 + d − d2 ≤ i ≤ d, if the site x was already visited before time n. Here we give an overview of this model when d1 + d2 = d and introduce and study the cases when d1 + d2 > d. In particular, we prove that for all the cases d ≥ 5 and most cases d = 4, the balanced excited random walk is transient.]
Published: Nov 4, 2020
Keywords: Excited random walk; Transience; Primary 60G50; 82C41; secondary 60G42
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