Short, Quantitative Construction of the IIC in Planar Percolation
Malo Hillairet
TL;DR
The paper proves a concise, self-contained convergence result for the Incipient Infinite Cluster in planar critical percolation by introducing a site-by-site coupling that samples conditioned configurations $\omega^{(m)}$ and $\omega^{(n)}$ above a common underlying percolation $\omega$. The main technical advance is revealing an outer black circuit to couple the conditioned configurations, which yields a quantitative bound in total variation to the IIC controlled by the dual one-arm probability, and in particular by the one-arm exponent $5/48$ up to subpolynomial factors. This provides the first explicit, explicit upper bound on the convergence speed and highlights a simple, robust method for obtaining convergence rates in planar models. The results connect classical transfer-matrix and scale-by-scale approaches with a transparent site-by-site exploration framework, offering a new perspective on the structure of conditional measures at criticality and opening questions about sharpness and higher-arm extensions.
Abstract
It is standard to construct the Incipient Infinite Cluster as the limit of the critical percolation measure conditioned on 0 being connected to radius n, as n tends to infinity. We provide a short proof of that convergence in the planar setting. A key step in the proof is to introduce an unbiased percolation configuration above which are coupled two percolations conditioned on 0 being connected to different radii. It implies a speed of convergence in total variation distance to the IIC measure upper-bounded by the dual one-arm probability, which is the first occurrence of an explicit upperbound.