P. Duncan & B. Schweinhart: Plaquette Percolation on the Torus

Abstract: 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 consider 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 $\varphi^*: H_i(P;\mathbb{Q})\to H_i(\mathbb{T}^d;\mathbb{Q})$ is the map on homology induced by the inclusion $\varphi: P\to \mathbb{T}^d$ , then $\mathbb{P}_p(\varphi^*\text{ is nontrivial})\to 0$ if $p<1/2$ and $\mathbb{P}_p(\varphi^* \text{ is nontrivial})\to 1$ if $p>1/2$ as $N\to\infty$ . 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.

In the first half of the talk, Paul will introduce the model, explain the relevant concepts from topology, and state a key topological duality result. In the second half, Ben will sketch a proof of the main result and describe related open questions for percolation on the torus and beyond. Joint work with Matthew Kahle (Ohio State).

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P. Duncan & B. Schweinhart: Plaquette Percolation on the Torus

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