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Graph distance and percolation dimension of three-dimensional Brownian motion
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 t
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