Multi-arm incipient infinite clusters in 2D: scaling limits and winding numbers

We study the alternating $k$-arm incipient infinite cluster (IIC) of site percolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman’s result that the scaling limit of critical site percolation on $\mathbb{T}$ is CLE${}_6$, we prove the existence of the scaling limit of the $k$-arm IIC for $k=1,2,4$. Conditioned on the event that there are open and closed arms connecting the origin to $\mathrm{\partial }{\mathbb{D}}_R$, we show that the winding number variance of the arms is $(3\mathrm{/}2+o(1))\log R$ as $R\to \mathrm{\infty }$, which confirms a prediction of Wieland and Wilson (2003). Our proof uses two-sided radial SLE${}_6$ and coupling argument. Using this result we get an explicit form for the CLT of the winding numbers, and get analogous result for the 2-arm IIC, thus improving our earlier result.