Discontinuity of the percolation density at the critical point

Determine whether the percolation density of the weight-dependent random connection model on the real line jumps discontinuously at its critical parameter \(\beta_c\).

Background

For one-dimensional independent long-range percolation at the critical decay, the percolation density is known to jump at the threshold. The paper asks whether an analogous discontinuity holds for the dependent weight-dependent random connection model studied here.

The numerical study suggests a jump of height close to $0.9$, but no proof is given.