Strange Random Topology of the Circle

Abstract Persistent homology of a point cloud is often computed to recover the topology of an underlying continuous geometric object. But this is only possible when the connectivity radius of the point cloud stays small. Could there be more to persistent homology, when the connectivity radius grows large? In this paper I bound probabilities that a random Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types.