Criticality at the boundary

Determine whether the weight-dependent random connection model on the real line with connection rule $(t\vee s)|x-y|\le\beta$ percolates at the boundary parameter \(\lambda\beta=1\).

Background

The paper proves that every connected component is finite when λβ<1\lambda\beta<1, while an infinite component exists when λβ≥31\lambda\beta\ge31. Thus, the exact behavior at the lower critical boundary λβ=1\lambda\beta=1 remains unresolved. The authors note that the corresponding classical independent-edge result includes equality, but their dependent-edge model does not.

At intensity one, this leaves the question of percolation at β=1\beta=1 open, despite the rigorous bound 1≤βc≤311\le\beta_c\le31.

References

The classical statement includes $\beta_{\mathrm{eff}=1$, whereas \Cref{thm:continuum} requires $\lambda\beta<1$, so the case $\lambda\beta=1$ is undecided.

— Rainbow percolation  (2608.12954 - Gracar et al., 13 Aug 2026) in Remark 2.1(v), Section 1 (Comparison with classical long-range percolation)

The true $\beta_c$ at intensity one is unknown beyond $\beta_c\in[1,31]$.

— Rainbow percolation  (2608.12954 - Gracar et al., 13 Aug 2026) in Remark 2.14(ii), Section 2.4 (A supercritical phase in the continuum)