Paul Duncan, Matthew Kahle, and Benjamin Schweinhart

We study plaquette percolation on a $d$-dimensional torus $\mathbb{T}^d_N$, defined by identifying opposite faces of the cube $[0,N]^d,$ and phase transitions marked by the appearance of giant (possibly singular) submanifolds that span the torus. 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 $\phi: P \hookrightarrow \mathbb{T}^d$, then

\[
\mathbb{P}_p\left(\text{$\phi_*$ is nontrivial}\right) \rightarrow
\begin{cases}
0 & \mbox{ if } p<\frac{1}{2} \\
1 & \mbox{ if } p>\frac{1}{2} \\
\end{cases}\]
as $N\rightarrow\infty.$ For the case $i=2$ and $d=4$ this implies that (possibly singular) surfaces spanning the $4$-torus appear abruptly at $p=1/2$. We also show that $1$-dimensional and $(d-1)$-dimensional plaquette percolation on the torus exhibit 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 analogous results for a site percolation model on the torus.

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https://cornell.zoom.us/j/93283820047?pwd=M29ETEtqL1h2cHhBM2xpeW5SRU4vUT09

Meeting ID: 932 8382 0047
Passcode: prob