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

TL;DR

This paper establishes the scaling limits and winding number properties of multi-arm incipient infinite clusters in 2D percolation, confirming theoretical predictions and advancing understanding of critical phenomena in lattice models.

Contribution

It proves the existence of scaling limits for 1, 2, and 4-arm IICs on the triangular lattice and derives the winding number variance, confirming prior predictions.

Findings

Winding number variance grows as (3/2) log R

Scaling limits exist for 1, 2, and 4-arm IICs

Explicit CLT for winding numbers

Abstract

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 $\partial \mathbb{D}_R$, we show that the winding number 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 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.