Limit distributions for topologies of random geometric complexes

A random geometric complex on a compact Riemannian manifold of dimension m is a union of balls centered at random points on the manifold — randomness comes from sampling the points from the uniform distribution. The radius r of these balls are chosen to scale together with the number n of sampled points: when $r=O(n^{-1/m})$ the geometric complex is in the thermodynamic regime. This is the regime when the topology of the complex is the richest. In this talk I will prove that when n goes to infinity the topological types of the components of this complex, appropriately rescaled, have a limit distribution. (Similar statements have recently been proved by Sarnak and Wigman for the topologies of nodal domains of random harmonics.) Of particular interest is the one-skeleton of this complex, which is a random geometric graph: in this case the existence of the previous limit implies the existence of a limit for the spectrum of this graph, which has interesting connections with the spectrum of the manifold.

(This is based on joint work with A. Auffinger, E. Lundberg)