Determine the exact walk dimension of the classical Sierpiński carpet

Determine the exact value of the walk dimension d_w for the classical Sierpiński carpet F.

Background

The paper uses the walk dimension d_w to describe sub-Gaussian heat-kernel estimates and the scaling of Dirichlet forms associated with Brownian motion on the Sierpiński carpet. Although bounds on d_w are known, the authors explicitly state that its exact value has not been determined even for the classical planar Sierpiński carpet. This remains an unresolved quantitative problem underlying the precise characterization of diffusion scaling on the carpet.