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Random planar curves and Schramm-Loewner evolutions. (English) Zbl 1057.60078
Tsirelson, Boris et al., Lectures on probability theory and statistics. Ecole d’Eté de probabilités de Saint-Flour XXXII – 2002. Berlin: Springer (ISBN 3-540-21316-3/pbk). Lecture Notes in Mathematics 1840, 109-195 (2004).
Between the years 2001 and 2003, Lawler, Schramm and Werner have published a series of ten striking joint papers on the application of complex analysis to the solution of difficult issues about the behaviour of paths of the planar Brownian curve and related questions. This course is a synthesis of their work. Starting from the discrete model of loop-erased random walks and considering their scaling limits they introduce the stochastic Löwner evolutions which are the stochastic counterpart of the deterministic Löwner chains that go back to 1923. This idea revealed itself to be very powerful to study Brownian intersection exponents, non intersection probabilities, critical percolation or uniform spanning trees. Their most spectacular result may be the rigorous proof of Mandelbrot’s conjecture on the dimension of the Brownian frontier. The course is a comprehensive survey of all these questions, with proofs sometimes only sketched.
For the entire collection see [Zbl 1045.60001].

MSC:
60J65 Brownian motion
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