About: Conformal loop ensemble

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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:

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  • 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. (en)
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  • 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: (en)
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  • Conformal loop ensemble (en)
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