Geometric Random Graphs on Circles

Given a dense countable set in metric space, the infinite random geometric graph is with given vertex and where any two points at distance less than 1 are connected, independently, some fixed probability. It has been observed by Bonato Janssen that some, but not all, such settings, resulting does depend on choices, sense it almost surely isomorphic to graph. While this notion makes general context of spaces, previous work restricted sets Banach spaces. We study case when underlying space circle circumference $L$, find surprising dependency behavior rationality $L$.