Plaquette Percolation on the Torus.

We study plaquette percolation on the torus $\mathbb{T}^d_N$, which is defined by identifying opposite faces of the cube $[0,N]^d$. The model we study starts with the complete $(i-1)$-dimensional skeleton of the cubical complex $\mathbb{T}^d_N$ and adds $i$-dimensional cubical plaquettes independently with probability $p$. Our main result is that if $d=2i$ is even and $\phi_*:H_{i}\left(P;\mathbb{Q}\right)\rightarrow H_{i}\left(\mathbb{T}^d;\mathbb{Q}\right)$ is the map on homology induced by the inclusion map $\phi: P \to \mathbb{T}^d$, then $\mathbb{P}_p\left(\text{$\phi_*$ is nontrivial}\right)\rightarrow 0$ if $p 1/2$ as $N\rightarrow\infty.$ We also show that $1$-dimensional and $(d-1)$-dimensional plaquette percolation on the torus have similar sharp thresholds at $\hat{p}_c$ and $1-\hat{p}_c$ respectively, where $\hat{p}_c$ is the critical threshold for bond percolation on $\mathbb{Z}^d,$ as well as bounds on critical probabilities in other dimensions.