Limit distributions for topologies of random geometric complexes

  • Antonio Lerario (SISSA, Trieste)
G3 10 (Lecture hall)


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)

Mirke Olschewski

MPI for Mathematics in the Sciences Contact via Mail

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