  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.
