Vacuum Cleaner Process     misc

Consider a rate-1 Poisson point process $\Pi$ in $\mathbb{R}^d$ ($d=2$ or $3$). The vacuum cleaner (or greedy walk) starts at the origin $x_0 = 0$ and moves as follows: at each step, it jumps to the nearest point of $\Pi$ that has not yet been visited, removes that point, and repeats.

Let $R(n) = |x_n|$ be the distance from the origin after $n$ steps. The process is a random walk in a random environment, and the growth rate of $R(n)$ depends on the dimension.

The simulation generates Poisson points lazily in unit cells as the walker explores, using a deterministic seed per cell for reproducibility and memory efficiency. Nearest-neighbor search uses grid-based spatial hashing with shell-by-shell expansion.