Connecting probability for random bounded-range one-dimensional network

We consider a class of bounded-range 1 D network models on a cycle and prove that, unlike the corresponding infinite-volume models, which never contain infinite components, they actually exhibit a phase transition for connectivity. We further show that depending on the specific choice of the edge probabilities, the last obstruction to connectivity can either be the existence of isolated vertices or the split of the cycle into two spatially separated components.