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≪N2 (energy is scarce), we show that there is an M-scale limit distribution related to a Darling–Mandelbrot law, while when M≫N2 (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where M/N2→ρ∈(0,∞), we show that there is an M-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.