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Energy-Constrained Random Walk with Boundary Replenishment

Wade, Andrew R.; Grinfeld, Michael

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Authors

Michael Grinfeld



Abstract

We study an energy-constrained random walker on a length-N interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of M on each boundary visit. We establish large N, M distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. Three phases are exhibited. When M≪N2 (energy is scarce), we show that there is an M-scale limit distribution related to a Darling–Mandelbrot law, while when M≫N2 (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where M/N2→ρ∈(0, ∞), we show that there is an M-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.

Journal Article Type Article
Acceptance Date Sep 11, 2023
Online Publication Date Oct 3, 2023
Publication Date 2023-10
Deposit Date Jul 4, 2023
Publicly Available Date Oct 3, 2023
Journal Journal of Statistical Physics
Print ISSN 0022-4715
Publisher Springer
Volume 190
Issue 10
Article Number 155
DOI https://doi.org/10.1007/s10955-023-03165-9
Keywords Energy and resource dynamics, 60G50, Darling–Mandelbrot distribution, 60J20, 92D40 (Secondary), 60J10 (Primary), Reflecting random walk, Metastability
Public URL https://durham-repository.worktribe.com/output/1170502

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This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.





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