Energy-constrained random walk with boundary replenishment

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 $M\mathrm{/}{N}^2\to \rho \in (0,\mathrm{\infty })$, we show that there is an $M$-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.