Exact critical value for one-dimensional long-range percolation

Determine the exact critical value C_c^{single}(σ) for percolation on a one-dimensional lattice in which pairs at distance r are connected independently with probability C r^{-(1+σ)}, for 0<σ<1.

Background

The paper uses one-dimensional long-range percolation as the single-layer baseline for its multiplex model. In this model, the edge probability between nodes at distance r is C/r{1+σ}, with 0<σ<1. An infinite connected cluster exists above a σ-dependent critical value C_c{single}(σ).

The paper states that only rigorous bounds and numerical estimates are available for this threshold, although the critical behavior has been established rigorously in closely related formulations. Determining an exact closed-form or otherwise exact characterization of C_c{single}(σ) would resolve the remaining threshold problem for the underlying single-layer model.

References

Despite its simplicity, the critical value of this model is not known exactly.

— Percolation on interdependent one-dimensional long-range networks  (2610.10097 - Sharabi et al., 7 Oct 2026) in Introduction, paragraph beginning “The simplest model of a network with LR links”

Closing the gap would require paths with steps in both directions. The sites visited by two such paths no longer form a renewal process, and we leave the proof that $2\zeta(1+\sigma)\,C_c(\sigma)\to1$ as an open problem.

— Explicit upper bounds on the threshold of one-dimensional long-range percolation  (2610.11872 - Amit, 8 Oct 2026) in Section Discussion, subsection “A factor of two”