Trivial, Critical and Near-critical Scaling Limits of Two-dimensional Percolation

TL;DR

This paper classifies the three fundamental types of scaling limits in two-dimensional percolation, linking them to how the percolation parameter approaches the critical point and illustrating their properties with elementary arguments.

Contribution

It provides a simple proof framework for the classification of scaling limits in 2D percolation, connecting them to the approach rate of the percolation parameter to criticality.

Findings

Identifies three types of scaling limits: trivial, critical, and near-critical.

Shows how the approach rate of p to p_c determines the limit type.

Demonstrates the persistence of macroscopic structures in near-critical limits.

Abstract

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, p_c, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to p_c. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to p_c. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.