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Conformal restriction and related questions. (English) Zbl 1189.60032
This survey gives an overview of the main results on the challenging subject of Schramm-Loewner evolution (SLE). After presenting the conformal invariance properties of Brownian motion, the author introduces a notion of restriction that allows one to compute scaling limits of measures on the set of paths conditioned to start and exit at specific points of the common boundary of simply connected domains by two different methods: either by conditioning or by conformal properties of the measures.
To introduce the notion of SLE the author proceeds by the intuition offered by self-avoiding walks. Although the results concerning self-avoiding walks are still at the level of conjectures (e.g., conjecture 1, p. 152), they are sufficiently convincing to motivate the introduction of SLE without needing a complete arsenal of difficult mathematical notions. By using non-rigorous physical arguments and very careful numerical simulations, physicists believe that self-avoiding walks of \(N\) steps have an end-to-end average distance asymptotically scaling like \(N^\nu\), with \(\nu=3/4\). In the physical literature, this is argued to imply that the continuum limit of the self-avoiding curve (if such a limit exists) has a Hausdorff dimension \(d_H=4/3\). (Notice, however, that \(\nu\) governs large-scale properties, while the Hausdorff dimension is a local property. Therefore, for the equality \(d_H=1/\nu\) to hold, an additional – plausible but unproved – assumption of stochastic scale invariance for self-avoiding paths is needed.) Now, theorem 2 states precisely that a chordal SLE with parameter \(8/3\) (denoted by \(\text{SLE}_{8/3}\)) is the unique probability on continuous paths without double points satisfying a conformal restriction, and moreover it is supported by curves of Hausdorff dimension \(4/3\). Conjecture 2, identifying limiting self-avoiding curves with SLEs of parameter \(8/3\), thus becomes highly appealing.
The subsequent more technical sections again use the motivation coming from self-avoiding walks and conformal invariance to define SLE via the Loewner equation. Two additional constructions of the SLE\(_{8/3}\) curve are introduced: through reflected Brownian motion and through a Poisson cloud of special Brownian excursions. Cases corresponding to a parameter different from \(8/3\) are also considered. Hand-waving interpretations of some martingales via non-intersection between independent samples of restriction measures are also given.
In summary, this beautiful survey gives the main results of this difficult topic and exposes the ideas behind their proofs, in a way which is nevertheless accessible to the non-specialist.

MSC:
60D05 Geometric probability and stochastic geometry
30C35 General theory of conformal mappings
60J65 Brownian motion
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
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