Homotopy distributions for Vietoris-Rips complexes of random circle samples

Abstract

We determine the distribution of the homotopy type of the Vietoris-Rips complex of a fixed number of independent uniform points on a circle. An exact identity expresses each binomial moment of a periodic-orbit count as one probability for circular gaps, multiplied by a constant avoidance factor. We evaluate these probabilities by finite lattice-point counts and interpolation, obtaining explicit formulas for every sample size and every real scale. On an explicit rational partition, the probabilities are continuous piecewise polynomials of degree at most one less than the sample size. We also determine the homotopy distribution in the first transition window above each positive critical fraction $c_t=t/(2t+1)$. In this window the number of critical periodic orbits converges to a Poisson law with explicit intensity, yielding the limiting probabilities of the preceding odd sphere, the contractible type, and every wedge multiplicity of the critical even sphere.