Many well-studied properties of random graphs have interesting generalizations to higher dimensions in random simplicial complexes. We will discuss a version of percolation in which, instead of edges, we add two (or higher)-dimensional cubes to a subcomplex of a large torus at random. In this setting, we see a phase transition that marks the appearance of giant "sheets," analogous to the appearance of giant components in graph models. This phenomenon is most naturally described in the language of algebraic topology, but this talk will not assume any topological background. 

Based on joint work with Ben Schweinhart and Matt Kahle.