Graph distance and percolation dimension of three-dimensional Brownian motion

Source 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 $0 0$.