Thematic Einstein Semester on
Geometric and Topological Structure of Materials
Summer Semester 2021
Speaker
Érika Roldán (TU München / EPFL)
Title
Topology of percolation clusters
Abstract
We study the topology of uniformly random and percolation finite clusters on regular tessellations of the Euclidian and Hyperbolic spaces. We prove, using a Pattern Theorem for clusters on locally finite regular lattices introduced by Madras in 1999, that the Betti numbers of these random clusters grow linearly with respect to the number of cells. We also sample large square two-dimensional percolation clusters using Markov Chain Monte Carlo Metropolis-Hasting algorithms. Based on these experiments, we conjecture that the expectation of the number of holes of a cluster attains its maximum at the site percolation threshold (that is approximately .5923) obeying also a law of large numbers as the number of cells goes to infinity. This is joint work with David Aristoff.
Contact
tes-summer2021@math.tu-berlin.de