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The covariant measure of SLE on the boundary. (English) Zbl 1244.60081

The authors study \(\gamma \cap \mathbb{R}\), the intersection of chordal SLE(\(\kappa\)) in \(\mathbb{H}\) with the real line, for \(4 <\kappa <8\): they construct a random measure \(\mu\) almost surely supported on \(\gamma \cap \mathbb{R}^+\), which gives additional information on the intersection, analogously to local times for the zeros of Brownian motion. This complements previous knowledge that \(\gamma \cap \mathbb{R}\) has Hausdorff dimension \(d=2-8/\kappa\), almost surely. The measure is essentially uniquely determined by a scaling and domain Markov property, together with a normalization and measurability / predictability property. The construction involves Doob-Meyer decomposition theory and a local martingale \(M_t(x)\) which may be interpreted as the conditional probability that the SLE curve will hit the point \(x\), having observed the curve up to time \(t\). The results transfer to boundaries of general smooth, simply connected domains. The article concludes by improving a convergence result of approximate measures \(\mu^\epsilon\) given by O. Schramm and W. Zhou [Probab. Theory Relat. Fields 146, No. 3-4, 435–450 (2010; Zbl 1227.60101)], showing that \(\mu^\epsilon\) converges weakly to the random measure \(\mu\) as \(\epsilon \searrow 0\). Finally the authors introduce a notion of “conformal Minkowski measure” which applies to \(\mu\), and conjecture that \(\mu\) is a Minkowski measure in the traditional sense.

MSC:

60J67 Stochastic (Schramm-)Loewner evolution (SLE)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60D05 Geometric probability and stochastic geometry
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics

Citations:

Zbl 1227.60101
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References:

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