Fractal dimensions of paths on three-dimensional random-walk traces
Abstract
We study four paths that join the starting site to the final position of a simple random walk of N=L2 steps on the infinite cubic lattice. The paths are selected on the walk trace---the graph of visited sites and traversed bonds---by different rules: the undirected shortest path S1, the directed shortest path S2, the time-respecting shortest path S3, and the depth De of the final site in a depth-first-search tree. Monte Carlo simulations for L up to 32768 give mean lengths scaling as LdO, with dS1=1.48005(9), dS2=1.49868(5), dS3=1.51205(5), and dDe=1.7099(7), so that dS1<dS2<dS3<dLE<dDe, where dLE=1.62400(5) is the known loop-erased-random-walk exponent. Imposing traversal directions increases the shortest-path exponent, and imposing time order increases it further; these constraints therefore change the scaling exponent, not only the amplitude. For every path, the standard deviation scales with the same exponent as the mean, so it is scale-invariant, unlike the visited-site mass; the rescaled length distributions collapse onto a finite-width scaling function. No analytical theory currently predicts these exponents; our results provide precise numerical benchmarks for future studies of path geometry on random-walk traces.
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BibTeX
@misc{shi2026fractal,
title = {{Fractal dimensions of paths on three-dimensional random-walk traces}},
author = {Qiyuan Shi and Xinyi Li and Youjin Deng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39612},
eprint = {2610.39612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39612},
}
RIS
TY - UNPB
TI - Fractal dimensions of paths on three-dimensional random-walk traces
AU - Qiyuan Shi
AU - Xinyi Li
AU - Youjin Deng
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39612
UR - https://arxiv.org/abs/2610.39612
AB - We study four paths that join the starting site to the final position of a simple random walk of N=L^2 steps on the infinite cubic lattice. The paths are selected on the walk trace---the graph of visited sites and traversed bonds---by different rules: the undirected shortest path S_1, the directed shortest path S_2, the time-respecting shortest path S_3, and the depth D_e of the final site in a depth-first-search tree. Monte Carlo simulations for L up to 32768 give mean lengths scaling as L^d_O, with d_S_1=1.48005(9), d_S_2=1.49868(5), d_S_3=1.51205(5), and d_D_e=1.7099(7), so that d_S_1<d_S_2<d_S_3<d_LE<d_D_e, where d_LE=1.62400(5) is the known loop-erased-random-walk exponent. Imposing traversal directions increases the shortest-path exponent, and imposing time order increases it further; these constraints therefore change the scaling exponent, not only the amplitude. For every path, the standard deviation scales with the same exponent as the mean, so it is scale-invariant, unlike the visited-site mass; the rescaled length distributions collapse onto a finite-width scaling function. No analytical theory currently predicts these exponents; our results provide precise numerical benchmarks for future studies of path geometry on random-walk traces.
ER -