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

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 $M \ll N^2$ (energy is scarce), we show that there is an $M$-scale limit distribution related to a Darling-Mandelbrot law, while when $M \gg N^2$ (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where $M / N^2 \to \rho \in (0,\infty)$, we show that there is an $M$-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.