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We study the alternating $k$-arm incipient infinite cluster (IIC) of sitepercolation on the triangular lattice $\mathbb{T}$. Using Camia and Newman'sresult that the scaling limit of critical site percolation on $\mathbb{T}$ isCLE$_6$, 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 armsconnecting the origin to $\partial \mathbb{D}_R$, we show that the windingnumber variance of the arms is $(3/2+o(1))\log R$ as $R\rightarrow \infty$,which confirms a prediction of Wieland and Wilson (2003). Our proof usestwo-sided radial SLE$_6$ and coupling argument. Using this result we get anexplicit form for the CLT of the winding numbers, and get analogous result forthe 2-arm IIC, thus improving our earlier result.

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