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Exploration trees and conformal loop ensembles. (English) Zbl 1170.60008
Two-dimensional statistical physics models often involve random collections of disjoint, non-intersecting loops in a planar lattice. The paper departs from a conjecture: when suitable boundary conditions are set in a random loop model on a simply connected planar domain, then as the grid size gets finer, the law of a random path connecting a pair of boundary points converges to the law of the chordal Schramm-Loewner evolution. The primary purpose of the paper is to investigate a family of candidate loop collections that have a scaling limit, called conformal loop ensembles. A secondary purpose is to formulate a series of conjectures and open questions for these ensembles, like e.g. about a continuum analog of the Fortuin-Kasteleyn cluster expansion for the Potts model, or scaling limits of the q-state Potts models. The focus is on hexagonal lattice graphs and so-called O(N) loop models. Bessel, skew Levy and Schramm-Loewner processes are considered. The radia Schramm-Loewner evolutions are found to be most natural candidates for the limiting laws of the exploration tree.

60D05 Geometric probability and stochastic geometry
82B27 Critical phenomena in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
05C80 Random graphs (graph-theoretic aspects)
60G52 Stable stochastic processes
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