Energy-Constrained Random Walk with Boundary Replenishment

Abstract

AbstractWe study an energy-constrained random walker on a length-Ninterval 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 Mon each boundary visit. We establish largeN, Mdistributional 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$$M \ll N^2$$M≪N2(energy is scarce), we show that there is anM-scale limit distribution related to a Darling–Mandelbrot law, while when$$M \gg N^2$$M≫N2(energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where$$M / N^2 \rightarrow \rho \in (0,\infty )$$M/N2→ρ∈(0,∞), we show that there is anM-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.