📝 SummarXiv

Abstract

It is standard to construct the Infinite Incipient 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.