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 T. Using Camia and Newman's result that the scaling limit of critical site percolation on T is CLE6, 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 partial derivative D-R, we show that the winding number variance of the arms is (3/2+ o(1)) log R as R -> 8, which confirms a prediction of Wieland and Wilson [Phys. Rev. E 68 (2003) 056101]. Our proof uses two-sided radial SLE6 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.