Conformal loop ensemble

These random collections of loops are indexed by a parameter κ, which may be any real number between 8/3 and 8.

[1] When 8/3 < κ ≤ 4, CLEκ may be alternatively constructed as the collection of outer boundaries of Brownian loop soup clusters.

Since CLEκ may be defined using an SLEκ process, CLE loops inherit many path properties from SLE.

[3] Since this dimension is strictly greater than 1+κ/8, there are almost surely points not contained in or surrounded by any loop.

However, since the gasket dimension is strictly less than 2, almost all points (with respect to area measure) are contained in the interior of a loop.

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 CLE 6 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 SLE 6 as the lattice spacing goes to zero.