The dimension of loop-erased random walk in 3D

TL;DR

This paper estimates the fractal dimension of 3D loop-erased random walk (LERW) as approximately 1.624, using simulations on different lattices, contributing precise numerical insights into LERW's geometric properties.

Contribution

The paper provides a highly precise numerical estimate of the fractal dimension of 3D LERW, improving understanding of its geometric complexity.

Findings

Estimated fractal dimension of 3D LERW as 1.62400 ± 0.00005

Face-centered cubic lattice shows slightly smaller corrections to scaling

Simulations on cubic and face-centered cubic lattices confirm the dimension estimate

Abstract

We measure the fractal dimension of loop-erased random walk (LERW) in 3 dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related to the uniform spanning tree and the abelian sandpile model. We simulated LERW on both the cubic and face-centered cubic lattices; the corrections to scaling are slightly smaller for the face-centered cubic lattice.