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Energy-Constrained Random Walk with Boundary Replenishment

Wade, Andrew R.; Grinfeld, Michael

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Michael Grinfeld


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
Keywords Energy and resource dynamics, 60G50, Darling–Mandelbrot distribution, 60J20, 92D40 (Secondary), 60J10 (Primary), Reflecting random walk, Metastability
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