Exploring an Infinite Space with Finite Memory Scouts

Consider a small number of scouts exploring the infinite d-dimensional grid with the aim of hitting a hidden target point. Each scout is controlled by a probabilistic finite automaton that determines its movement (to a neighboring grid point) based on its current state. The scouts, that operate under a fully synchronous schedule, communicate with each other (in a way that affects their respective states) when they share the same grid point and operate independently otherwise. Our main research question is: How many scouts are required to guarantee that the target admits a finite mean hitting time? Recently, it was shown that d+ 1 is an upper bound on the answer to this question for any dimension d ≥ 1 and the main contribution of this paper comes in the form of proving that this bound is tight for d ∈ {1, 2}.