Percolation on Hypergraphs with Four-Edges

We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities $p$, $r$, and $t$ connecting the four vertices. We find explicit values of these probabilities that satisfy the self-duality conditions discussed by Bollob\'as and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions ($p$, $r$, $t$). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold $p = 0.441374\pm 0.000001$, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.