  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".
