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 rightarrow 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.
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