 Solution

Put $s=1-p$. On the four-cycle, the total unnormalized weight isThe event $N\leftrightarrow S$ contains all configurations with zero or one closed edge and exactly two of the six configurations with two closed edges, so its weight isDivide numerator and denominator by $q$. As $s\to 0$ and $sq\to \infty$, the omitted numerator terms are $o(1+2s^{2}q)$, while the middle denominator terms areConsequently