The Scaling Limit Geometry of Near-Critical 2D Percolation

Abstract

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+λδ^{1/ν}$, with $ν=4/3$, as the lattice spacing $δ\to 0$. Our proposed framework extends previous analyses for $p=p_c$, based on $SLE_6$. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as $λ$ varies. This structure is rich enough to yield a one-parameter family of near-critical loop processes and their associated connectivity probabilities as well as related processes describing, e.g., the scaling limit of 2D minimal spanning trees.