Starving Random Walks

TL;DR

This paper reviews recent findings on the starving random walk model, analyzing how resource-limited exploration affects the walker's lifetime and food collection based on inter-visit times and starvation thresholds.

Contribution

It introduces a framework to quantify the lifetime and food collection of starving random walks using inter-visit times and maximum inter-visit time analysis.

Findings

Characterization of inter-visit times and their maxima.

Relation between maximum inter-visit time and starvation lifetime.

Quantitative estimates of food units collected before starvation.

Abstract

In this chapter, we review recent results on the starving random walk (RW) problem, a minimal model for resource-limited exploration. Initially, each lattice site contains a single food unit, which is consumed upon visitation by the RW. The RW starves whenever it has not found any food unit within the previous $\mathcal{S}$ steps. To address this problem, the key observable corresponds to the inter-visit time $\tau_k$ defined as the time elapsed between the finding of the $k^\text{th}$ and the $(k+1)^\text{th}$ food unit. By characterizing the maximum $M_n$ of the inter-visit times $\tau_0,\dots,\tau_{n-1}$, we will see how to obtain the number $N_\mathcal{S}$ of food units collected at starvation, as well as the lifetime $T_\mathcal{S}$ of the starving RW.