Consider critical site percolation on the planar triangular lattice. Denote by $A_j(m,n)$ the event that there are $j$ arms (paths from the inner boundary to the outer boundary) of alternating colour within the annulus $\Lambda_{m,n} := [-n,n]^2\backslash (-m,m)^2$, and $A_j^+(m,n)$ the event that there are $j$ arms of alternating colour within the half-annulus $\Lambda_{m,n} \cap \mathbb{H}$.
I am trying to follow Wendelin Werner's notes on critical planar percolation. The first exercise sheet gives elementary arguments for the values of the boundary three-arm and whole-plane five-arm exponent: \begin{equation} \mathbb{P}(A_3^+(1,n)) \asymp \frac{1}{n^2}, \end{equation} \begin{equation} \mathbb{P}(A_5(1,n)) \asymp \frac{1}{n^2}. \end{equation}
Later, in Section 6.3, Werner states that the more general, multi-scale, estimates for the three and five arm events can be derived "using the very same arguments as in the exercise sheet": \begin{equation} \mathbb{P}(A_3^+(m,n)) \leq \left ( \frac{m}{n} \right )^2, \end{equation} \begin{equation} \mathbb{P}(A_5(m,n)) \leq \left ( \frac{m}{n} \right )^2. \end{equation}
However, I find myself unable to generalise these arguments; the arguments for the upper bounds rely on counting vertices which satisfy certain 'geometric features,' which I cannot see working when counting 'blocks' of vertices instead. Moreover, other 'multi-scale' bounds for other arm events that I have come across elsewhere require non-elementary or very technical lemmas, such as the arm separation lemma.
I would appreciate if anyone would be able to provide a more in-depth explanation of how we can obtain these multi-scale estimates using elementary arguments, such as RSW estimates, locality or the FKG inequality. BK inequality is fine too, though it would be enlightening to see if these can be proved without it.
