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Graph distance and percolation dimension of three-dimensional Brownian motion
Abstract
We prove that the minimum Hausdorff dimension of a continuous curve joining W0 to W1 within the range W([0,1]) of a three-dimensional Brownian motion is almost surely a deterministic constant γ3∈(1,5/3]. Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension γ3 and admits a parametrization that is β-Hölder for every 0<β<1/γ3. To identify γ3, we partition Brownian time into n equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent θ and that γ3=2θ. We also establish npθ+o(1) growth for their p-th moments, for every p>0.
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BibTeX
@misc{shiraishi2026graph,
title = {{Graph distance and percolation dimension of three-dimensional Brownian motion}},
author = {Daisuke Shiraishi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32137},
eprint = {2610.32137},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32137},
}
RIS
TY - UNPB
TI - Graph distance and percolation dimension of three-dimensional Brownian motion
AU - Daisuke Shiraishi
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32137
UR - https://arxiv.org/abs/2610.32137
AB - We prove that the minimum Hausdorff dimension of a continuous curve joining W_0 to W_1 within the range W([0,1]) of a three-dimensional Brownian motion is almost surely a deterministic constant γ_3∈(1,5/3]. Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension γ_3 and admits a parametrization that is β-Hölder for every 0<β<1/γ_3. To identify γ_3, we partition Brownian time into n equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent θ and that γ_3=2θ. We also establish n^pθ+o(1) growth for their p-th moments, for every p>0.
ER -