Graph distance and percolation dimension of three-dimensional Brownian motion

Abstract

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 $γ_3\in(1,5/3]$. 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 $p>0$.