number of closed-loop graphs in square lattice

Question

Suppose I have a $N\times N$ square lattice. I want to know how many different closed-loop diagrams are there. The closed-loop diagrams can have no loop, one loop, two loops, etc. If two loops share the same boundary, then the boundary disappears and these two loops merge into one big loop. Next, I want to know how many different closed-loop diagrams are there going through a particular link. If the ratio of number of different closed-loop diagrams through a particular link to the number of different closed-loop diagrams a fixed number? Furthermore, how many different closed-loop diagrams are there going through two particular links and is the ratio to the total number of different diagrams a fixed number?

Answer 1

The number of closed loop diagrams is equal to the number of subsets of the set of small squares. It is clear that any such subset results in a closed-loop diagram by cancelling common edges. On the other hand given a closed-loop diagram, for each loop, mark all the small squares that lie within that loop. A square lying inside more than one loop is marked multiple times. The set of squares that have been marked an odd number of times generates the loop diagram. Therefore, there are $2^{N^2}$ closed-loop diagrams.

If a link lies on the edge of the large square it will, will be in a closed loop diagram once for every subset of the other squares, so in $2^{N^2-1}$ closed-loop diagrams, that is, half the total.

All other edges lie on two squares. They lie on the diagram is one or the other or both squares are in the subset, along with any subset of the other squares. This is $2\cdot2^{N^2-2}$, so half the total once again.