Consider a two dimensional square lattice and let $C_n$ be the number of self avoiding cycles (i.e., a self avoiding walk that starts at the origin and terminates at the origin) such that the number of vertices ``inside'' the cycle is $n$.
Does $C_n$ grow exponentially with n?
More precisely, by ``number of vertices inside the cycle'', we mean the number of vertices which are in the set whose boundary is the cycle and we do not count the vertices belonging to the cycle itself.
Comment. If $C_n$ was the number of cycles having $n$ vertices, the behaviour of $C_n$ would be known to be as $\sim \mu^n$, where $\mu$ is a constant depending only on the lattice, known as connective constant. This comes from a subadditive argument.