The Scaling Limit Geometry of Near-Critical 2D Percolation

TL;DR

This paper investigates the geometric structure of near-critical 2D percolation in the scaling limit, extending critical models with new probabilistic tools to understand connectivity and related processes.

Contribution

It introduces a novel framework combining loop processes and Poissonian marking to analyze near-critical percolation beyond the critical point.

Findings

Develops a continuum model for near-critical 2D percolation

Provides a one-parameter family of loop processes and connectivity probabilities

Connects the scaling limit to minimal spanning trees in 2D

Abstract

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing $\delta \to 0$. 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 $\lambda$ 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.