Alladi Ramakrishnan Hall

Central limit theorems for some random simplicial complexes

D. Yogeswaran

Technion, Israel

In this talk, we shall state central limit theorems for many local
and global functionals of simplicial complexes built on various random point
processes. In the first part of the talk we will consider simplicial counts
in Cech and Voronoi simplicial complexes for long-range dependent point
processes such as zeros of Gaussian analytic functions and determinantal
point processes. These functionals serve as a good illustration
of our general central limit theorems for local functionals of the above
point processes.

In the second part, we shall restrict ourselves to the ubiquitous Poisson
point process but look at a very global functional called the Betti number.
Apart from proving normal convergence, for certain regimes, we shall be able
to give 'optimal' rates of convergences as well. We shall show various
stabilizing properties of the Betti numbers of the random Cech complex to
leverage recent results on stabilizing functionals of Poisson point
processes. Time permitting, we shall hint at strong laws or Poisson limit
theorems for some of these functionals.

The necessary notions of simplicial complexes and point processes shall be
defined in the talk. The talk is based on recent research projects with
R.J.Adler, E.Subag, B.Blaszczyszyn and J.E.Yukich.