The scaling limits of near-critical and dynamical percolation

TL;DR

This paper establishes the existence of scaling limits for near-critical and dynamical percolation on the triangular lattice, demonstrating their conformal covariance and Markovian properties in the quad-crossing space.

Contribution

It introduces the first rigorous proof of scaling limits for near-critical and dynamical percolation, extending the critical model's limit using pivotal measures.

Findings

Scaling limits exist for near-critical and dynamical percolation

The limits are Markovian and conformally covariant

The approach uses perturbations of the critical model's limit

Abstract

We prove that near-critical percolation and dynamical percolation on the triangular lattice $\eta \mathbb{T}$ have a scaling limit as the mesh $\eta \to 0$, in the "quad-crossing" space $\mathcal{H}$ of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.