Determine whether unrectifiable directed fractal percolation occurs in low dimensions

Determine whether unrectifiable directed fractal percolation occurs in dimensions two and higher but below the sufficiently large dimensions covered by the high-dimensional construction, including whether suitable anisotropic box side lengths can produce directed connectivity.

Background

The paper proves that an anisotropic fractal percolation model with spatial scaling factor 8 and temporal scaling factor 10 exhibits directed percolation when the spatial dimension is sufficiently large. The result does not determine the lowest dimension in which this phenomenon occurs. The authors note that planar models with rectangular boxes of arbitrary fixed integer side-scaling factors still fail to exhibit directed connectivity, while suggesting that dimensions three and higher may permit constructions with appropriately chosen aspect ratios.

References

Theorem \ref{t:8x10} leaves open the intriguing question of determining whether unrectifiable directed fractal percolation occurs in low dimensions.

— Directed fractal percolation and non-Lipschitz variants  (2610.12366 - Ganguly et al., 8 Oct 2026) in Remark following Theorem 8x10, Section 1, subsection “Unrectifiable directed fractal percolation”