Rainbow percolation

We consider the weight-dependent random connection model on a Poisson point process of intensity $λ$ on $\mathbb{R}\times(0,1)$ in which the vertices $(x,t)$ and $(y,s)$ are joined precisely when $(t\vee s)|x-y|\leqβ$. Points at distance $d$ are joined with probability $\min(1,β/d)^2$, the critical decay of one-dimensional long-range percolation, and edges sharing a vertex are dependent through the common mark. We prove that the model has a genuine phase transition: for $λβ<1$ almost surely all connected components are finite, while for $λβ\geq 31$ an infinite component exists, so at intensity one the critical value satisfies $β_c\in[1,31]$; a numerical study included as an appendix places it near $2$. The lower bound is proved via a discrete skeleton of the model, obtained by pinning the vertices to $\mathbb{Z}$, which is of independent interest: it has no supercritical phase at all, jumping at a degenerate transition from total fragmentation to trivial connectivity, even though almost surely infinitely many edges cross every fixed site.