Critical-parameter question for planar soft-stick percolation

Consider the directed random graph on $\mathbb{Z}^2$ in which each vertex independently chooses one of its four nearest-neighbour directions uniformly, places an arrow in that direction, and independently places an arrow in the opposite direction with probability $\epsilon\in[0,1]$ . Let $C^+_\epsilon(0)$ be the forward cluster of the origin, consisting of vertices reachable from $0$ by following arrows, and define $\epsilon_c:=\inf\{\epsilon\in[0,1]:\mathbb{P}(|C^+_\epsilon(0)|=\infty)>0\}$ . Determine the exact value of $\epsilon_c$ .

References

Primary source

arXiv

Additional references

A Peierls bound for planar soft-stick percolation — arXiv

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A new argument shows the transition occurs below the model’s two-arrow endpoint, but the exact transition value remains unknown.

The problem asks for the exact critical parameter governing percolation in planar soft-stick percolation. The new result gives a strict bound but does not determine that parameter.

September 2026 Peierls bound

A transfer-matrix Peierls argument establishes percolation strictly below the two-arrow endpoint, yielding explicit progress toward the critical parameter. The result is reported in a preprint; its mathematical correctness has not been independently verified.

Current status (as of September 2026): Percolation is claimed below the two-arrow endpoint, while the exact critical parameter remains open.

Sources

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