Conformal loop ensemble
A conformal loop ensemble 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.
Constructions
For 8/3 < κ < 8, CLEκ may be constructed using a branching variation of an SLEκ process. When 8/3 < κ ≤ 4, CLEκ may be alternatively constructed as the collection of outer boundaries of Brownian loop soup clusters.Properties
CLEκ is conformally invariant, which means that if is a conformal map, then the law of a CLE in D' is the same as the law of the image of all the CLE loops in D under the map.Since CLEκ may be defined using an SLEκ process, CLE loops inherit many path properties from SLE. For example, each CLEκ loop is a fractal with almost-sure Hausdorff dimension 1 + κ/8. Each loop is almost surely simple when 8/3 < κ ≤ 4 and almost surely self-touching when 4 < κ < 8.
The set of all points not surrounded by any loop, which is called the gasket, has Hausdorff dimension 1 + 2/κ + 3κ/32 almost surely are contained in the interior of a loop.
CLE is sometimes defined to include only the outermost loops, so that the collection of loops is non-nested. Such a CLE is called a simple CLE to distinguish it from a full or nested CLE. The law of a full CLE can be recovered from the law of a simple CLE as follows. Sample a collection of simple CLE loops, and inside each loop sample another collection of simple CLE loops. Infinitely many iterations of this procedure gives a full CLE.