The celebrated Harris–Kesten theorem is that the critical probability for bond percolation in the square lattice is 1/2. It has been a folklore problem for apparently some time to find an appropriate analogue of this theorem in high dimensions.

Duncan, Schweinhart, and I studied "plaquette percolation" in a d-dimensional torus made by identifying opposite sides of a subdivided d-dimensional cube, where k-dimensional cubical cells are inserted independently with probability p. For an appropriate homological definition of "giant cycles", and whenever d=2k, we show that the critical probability is indeed 1/2.

Since this is a talk in probability seminar, I will aim to focus on the topological definitions and main ideas. The talk will be mostly self-contained.