I'm reading Creswick, Farach, and Poole's book called Introduction to Renormalization Group Methods in Physics (unfortunately, it's out of print). Despite the book being about physics, I was hoping for some clarification on something known as the "matching lattice".
In Chapter 3, there is a discussion about percolation, both of the site and bond types. I know that you can construct a dual lattice by placing a vertex in the middle of each elementary "face" of the lattice and creating bonds that are orthogonal to the bonds in the original lattice. In this way, you can show that the square lattice is self-dual.
However, the authors then talk about a "matching lattice", which is defined as such:
"The matching lattice of $L$ is constructed by completing each elementary polygon or plaquette of L, that is, connecting each vertex of the polygon by bonds and replacing completed polygons by the corresponding plaquette."
I think I see how this works for a square lattice, but my issue is that the authors claim that the triangular lattice is self-matching, because all of its elementary polygons are already triangles. I just can't seem to connect this to the matching lattice. Am I overthinking things?
Is the matching lattice for a triangular lattice the exact same lattice because you cannot make any new connections between the vertices? I am trying to study site percolation (where each vertex in the lattice is either filled or vacant), and I was hoping to explicitly construct the matching lattice. Maybe I'm struggling because this is a case which is sort of degenerate. Any clarification would be appreciated!