Consider critical percolation in two dimensions. Under the condition that there are k disjoint alternating open and closed arms crossing the annulus $A(l,n)$, we prove a central limit theorem and variance estimates for the winding angles of the arms (as $n\rightarrow \infty$, $l$ fixed). This result confirms a prediction of Beffara and Nolin (Ann. Probab. 39: 1286-1304, 2011). Using this theorem, we also get a CLT for the multiple-armed incipient infinite cluster (IIC) measures.
"A CLT for winding angles of the arms for critical planar percolation." Electron. J. Probab. 18 1 - 20, 2013. https://doi.org/10.1214/EJP.v18-2285