Conformal loop ensemble

A conformal loop ensemble (CLE_κ) 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. CLE_κ is a loop version of the Schramm-Loewner evolution: SLE_κ is designed to model a single discrete random interface, while CLE_κ models a full collection of interfaces.

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

For 8/3 < κ < 8, CLE_κ may be constructed using a branching variation of an SLE_κ process (Sheffield (2009)). When 8/3 < κ ≤ 4, CLE_κ may be alternatively constructed as the collection of outer boundaries of Brownian loop soup clusters (Sheffield and Werner (2010)).

CLE_κ is conformally invariant, which means that if ${\displaystyle \varphi :D\to D'}$ 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 ${\displaystyle \varphi }$.

Since CLE_κ may be defined using an SLE_κ process, CLE loops inherit many path properties from SLE. For example, each CLE_κ loop is a fractal with almost-sure Hausdorff dimension 1+κ/8. Each loop is almost surely simple (no self intersections) when 8/3 < κ ≤ 4 and almost surely self-touching when 4 < κ < 8.

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