It was established by Dennis, Kitaev et al. that the 2D Toric Code can be mapped to a 2D Random Bond Ising Model. The original derivation was given in the paper "Topological quantum memory" which can be found in J. Math. Phys. 43, 4452 (2002); doi: 10.1063/1.1499754 or online at http://dx.doi.org/10.1063/1.1499754 . A free arXiv preprint version is available online here.
The derivation that shows that the 2D Toric Code can actually mapped to a 2D Random Bond Ising Model is given in section IV - D "Derivation of the model" (Starting page 4469).
The point where I'm getting lost is when he is trying to set up a function that for a fixed chain E outputs the probability of a "homotopically equivalent" chain $E'$ (which he calls $p(E'|E)$).
To obtain this function he first calculates the probability of a link being occupied that lies on $E'$ but not on $E$. He obtains (Eq. 13) that the probability is equal to
\begin{equation} \left(\frac{p}{1-p}\right)^{n_{C}(\ell)} \end{equation}
up to an overall normalization and with $C$ being the cycle corresponding to $E'$.
I don't really see how he arrives at that results or what the overall normalization factor actually is (after all "normalization" is crucial when talking about probabilities - what is the meaning of probability 35 if I don't know the normalization?) Then he obtains in Eq. (15) an explicit exponential-like form for $p(E'|E)$ which isn't clear to me neither. He also seems to omit a lot of explanations at least at this stage.
Could anyone explain to me in a bit more details how the derivation is carried out in more detail? This would really help me a lot.