Conformal loop ensemble

In critical percolation on the honeycomb lattice, each hexagon face is colored red or black independently with equal probability. Every interface separating a black cluster from a red cluster is shown in green. This random collection of interfaces converges in law to CLE6 as the lattice spacing goes to zero.
To define a random interface converging to SLE, we fix the colors of the hexagons along the boundary of the domain. This procedure defines a single interface separating red hexagons from black hexagons. This path converges in law to SLE6 as the lattice spacing goes to zero.

A conformal loop ensemble (CLEκ) is a random collection of non-crossing loops in a simply connected, open subset of the plane. These random collections of loops are indexed by a parameter κ, which may be any real number between 8/3 and 8. CLEκ is a loop version of the Schramm–Loewner evolution: SLEκ is designed to model a single discrete random interface, while CLEκ models a full collection of interfaces.

In many instances for which there is a conjectured or proved relationship between a discrete model and SLEκ, there is also a conjectured or proved relationship with CLEκ. For example:

  • CLE3 is the limit of interfaces for the critical Ising model.
  • CLE4 may be viewed as the 0-set of the Gaussian free field.
  • CLE16/3 is a scaling limit of cluster interfaces in critical FK Ising percolation.
  • CLE6 is a scaling limit of critical percolation on the triangular lattice.