Random geometric complexes in the thermodynamic regime

Understand

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices.

• In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems.
• As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called `thermodynamic' regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers.
• The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer-Vietoris exact sequences.