Energy-constrained random walk with boundary replenishment

TL;DR

This paper analyzes the lifetime of an energy-constrained random walk on a one-dimensional lattice with boundary replenishment, revealing three distinct asymptotic phases depending on the energy capacity relative to the interval size.

Contribution

It provides the first large-scale asymptotic analysis of the walker's lifetime, identifying phase transitions and deriving explicit limit distributions for different energy regimes.

Findings

For scarce energy ($M \,\ll\, N^2$), the lifetime follows a Darling-Mandelbrot law.

For plentiful energy ($M \,\gg\, N^2$), the lifetime distribution is exponential on a stretched-exponential scale.

At the critical scaling ($M / N^2 \to \rho$), the lifetime converges to an infinitely-divisible distribution involving theta functions.

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