# percolation - number of cycles around the origin

I try to study Percolation Theory by "A mini course on percolation theory" of Jeffrey E. Steif.

I am very curios about Exercise 2.4.

Show that the number of cycles around the origin of length n is at most $n4(3^{n−1})$.

I need to this on lattice and it's dual representation.

In such cases I often try to reverseengineer the formula, in this particular case: 4 may stand for four directions, $n$ may be initial choice for the first edge, and $3^{n-1}$ maybe just continuation in all 3 least directions after the initial one was chosen.

The question is why do we need $4n$, why $n3^{n-1}$ is not sufficient.

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This is a crude estimate, which could be refined but such refinements are useless in this context. The factor $n$ may enumerate the vertex on the cycle on the horizontal halfline $\mathbb N\times\{0\}$ passing through the origin, closest to the origin on its right (then the first factor 4 might probably be replaced by 3). –  Did Nov 8 '12 at 10:51