Energy-constrained random walk with boundary replenishment

Andrew Wade and Michael Grinfeld

Journal of Statistical Physics, 190, October 2023, article no. 155. DOI: 10.1007/s10955-023-03165-9 [Article] [arXiv] [MR]

Supported by EPSRC award Anomalous diffusion via self-interaction and reflection (EP/W00657X/1).



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 MN2 (energy is scarce), we show that there is an M-scale limit distribution related to a Darling-Mandelbrot law, while when MN2 (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where M/N2ρ(0,), we show that there is an M-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.