Random planar curves and Schramm-Loewner evolutions

Details Random planar curves and Schramm-Loewner evolutions Random planar curves and Schramm-Loewner evolutions

2003-03-27

arxiv.org/abs/math/0303354v1

Mathematics

Probability

(Final?) version of the Notes of the lectures delivered at the Saint-Flour summer school (July 2002). 100 pages, 16 figures. T

Scientific paper

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the intuitive ideas that lead to the definition of the Schramm-Loewner evolutions SLE, we define these objects, study its various properties, show how to compute (probabilities, critical exponents) using SLE, relate SLE to planar Brownian motions (i.e. the determination of the critical exponents), planar self-avoiding walks, critical percolation, loop-erased random walks and uniform spanning trees.

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