Homological percolation on a torus: Plaquettes and permutohedra

We study higher-dimensional homological analogues of bond percolation on a square lattice and site percolation on a triangular lattice. By taking a quotient of certain infinite cell complexes by growing sublattices, we obtain finite cell complexes with a high degree of symmetry and with the topology of the torus Tᵈ. When random subcomplexes induce nontrivial i-dimensional cycles in the homology of the ambient torus, we call such cycles giant. We show that for every i and d there is a sharp transition from nonexistence of giant cycles to giant cycles spanning the homology of the torus. We also prove convergence of the threshold function to a constant in certain cases. In particular, we prove that pc=1/2 in the case of middle dimension i=d/2 for both models. This gives finite-volume high-dimensional analogues of Kesten's theorems that pc=1/2 for bond percolation on a square lattice and site percolation on a triangular lattice.