Short, Quantitative Construction of the IIC in Planar Percolation

Abstract

The classical definitions of the Incipient Infinite Cluster (IIC) of percolation consist in conditioning the origin on being connected to radius $n$ and letting $n$ go 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 that the speed of convergence in total variation distance to the IIC measure is upper-bounded by the dual one-arm probability, which is the first occurrence of an explicit upper-bound. Additionally, the proof can be generalised widely to planar graphs. For example, under the assumption that there is no infinite dual cluster at criticality, it proves existence of the IIC on any planar triangulation.