Conformal Loop Ensembles: The Markovian Characterization

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Abstract
For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural property that is conjecturally satisfied by the scaling limits of interfaces in various models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears at first glance to be weaker. Our main result is that there exists at most a one-dimensional family of random loop collections with this property---one for each k in (8/3,4] - and that the loops are necessarily forms of SLE(k). In a second paper, we will show (using loop-soups) that these random loop collections actually exist for all k in (8/3,4].

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