SLE₆ and CLE₆ from critical percolation

Summary

We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE₆ and the full scaling limit of cluster interface loops. The results given here on the full scaling limit and its conformal invariance extend those presented previously. For site percolation on the triangular lattice, the results are fully rigorous. We explain some of the main ideas, skipping most technical details.

1. Introduction

In the theory of critical phenomena it is usually assumed that a physical system near a continuous phase transition is characterized by a single length scale (the correlation length) in terms of which all other lengths should be measured. When combined with the experimental observation that the correlation length diverges at the phase transition, this simple but strong assumption, known as the scaling hypothesis, leads to the belief that at criticality the system has no characteristic length, and is therefore invariant under scale transformations. This suggests that all thermodynamic functions at criticality are homogeneous functions, and predicts the appearance of power laws.

It also implies that if one rescales appropriately a critical lattice model, shrinking the lattice spacing to zero, it should be possible to obtain a continuum model, known as the scaling limit. The scaling limit is not restricted to a lattice and may possess more symmetries than the original model. Indeed, the scaling limits of many critical lattice models are believed to be conformally invariant and to correspond to Conformal Field Theories (CFTs).