Conformal loop ensemble

For 8/3 < κ < 8, CLE_κ may be constructed using a branching variation of an SLE_κ process.^[1] When 8/3 < κ ≤ 4, CLE_κ may be alternatively constructed as the collection of outer boundaries of Brownian loop soup clusters.^[2]

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.

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 (random soups, carpets and fractal dimensions by Nacu and Werner.^[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.

CLE is sometimes defined to include only the outermost loops, so that the collection of loops is non-nested (no loop is contained in another). 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.