On the size of earthworm's trail
Probability
2023-10-03 v2
Abstract
We investigate the number of holes created by an ``earthworm'' moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least after steps.
Cite
@article{arxiv.2303.02742,
title = {On the size of earthworm's trail},
author = {Krzysztof Burdzy and Shi Feng and Daisuke Shiraishi},
journal= {arXiv preprint arXiv:2303.02742},
year = {2023}
}