Energy-Constrained Random Walk with Boundary Replenishment

Wade, Andrew R.; Grinfeld, Michael

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.

Citation

Wade, A. R., & Grinfeld, M. (2023). Energy-Constrained Random Walk with Boundary Replenishment. Journal of Statistical Physics, 190(10), Article 155. https://doi.org/10.1007/s10955-023-03165-9

• 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
• Electronic ISSN: 1572-9613
• 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