Energy-constrained random walk with boundary replenishment

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.

arXiv comment32 pages, 1 figure; v2: minor revisions, some additional exposition

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