Thematic Einstein Semester on
Geometric and Topological Structure of Materials
Summer Semester 2021
Speaker
Benjamin Schweinhart (George Mason U)
Title
Plaquette Percolation on the Torus
Abstract
We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. The former model – plaquette percolation – is defined on the d-dimensional torus TdN obtained by identifying opposite faces of the cube [0,N]d. It is the random i-complex P which contains the complete (i-1)-dimensional skeleton of the cubical complex TdN and each i-dimensional cubical plaquette is added independently with probability p. Our main result is that if d=2i is even and Φ*: Hi(P;ℚ) → Hi(Td;ℚ) is the map on homology induced by the inclusion Φ: P → Td, then ℙp( Φ* is nontrivial) → 0 if p<1/2 and ℙp(Φ* is surjective) → 1 if p>1/2 as N → ∞. We also show the existence of a sharp threshold function for other values of i and d, as well as analogous results for the site percolation model.
(Joint work with Paul Duncan and Matthew Kahle.)
Contact
tes-summer2021@math.tu-berlin.de