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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