Strange Random Topology of the Circle

Abstract

We characterise high-dimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odd-dimensional sphere or a bouquet of even-dimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".