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Random curves, scaling limits and Loewner evolutions. (English) Zbl 1393.60016

This paper deals with Schramm’s SLE curves. An \(\mathrm{SLE}_k\), \(k>0\) is a random collection of hulls \((K_t)_{t\geq0}\) corresponding to a random driving function \(W_t=\sqrt k\cdot B_t\) where \((B_t)_{t\geq0}\) is a standard one dimensional Brownian motion.
The authors investigate sequences of random planar curves and establish sufficient conditions for their precompactness.
The main result of this article involves the Loewner equation, i.e., curves that are scaling limits of sequences of simple curves.

MSC:

60D05 Geometric probability and stochastic geometry
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
30C35 General theory of conformal mappings
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