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Basic properties of SLE. (English) Zbl 1081.60069
The authors study the so called stochastic Loewner evolution (SLE) which is simply a random growth process defined as follows: let $$B_{t}$$ be a Brownian motion on $$\mathbb{R}$$, started from $$B_{0}=0$$; for $$\kappa\geq 0$$ let $$\xi(t)=\sqrt{\kappa}B(t)$$ and for each $$z\in\overline{\mathcal{H}}/{0}$$ (where $$\overline{\mathcal{H}}$$ is the closed upper half plane) let $$g_{t}(z)$$ be the solution of the ordinary (stochastic!) differential equation $\partial_{t}g_{t}(z) =\frac{2}{g_{t}(z) -\xi(t)},\quad g_{0}(z)= z.$ The parametrized collection of maps $$\{g_{t}: t\geq 0\}$$ called the chordal $$\text{SLE}_{\kappa}$$ is the central object of study by the authors. The trace $$\gamma$$ of $$\text{SLE}$$ is defined by $$\gamma(t) = \lim_{z\to 0}\hat{f}_{t}(z)$$ where $$f_{t}=g_{t}^{-1},\; \hat{f}_{t}(z)=f_{t}(z+\xi(t))$$. The authors build up an elaborate set of analytical tools to establish that the trace is a simple path for $$\kappa\in[0,4]$$, a self intersecting path for $$\kappa\in (4,8)$$ and for $$\kappa >8$$, it is a space filling. The authors also establish that the Hausdorff dimension of $$\text{SLE}_{\kappa}$$ trace is almost surely at most $$1+\kappa/8$$ and that the expected number of disks of size $$\varepsilon$$ needed to cover it inside a bounded set is at least $$\varepsilon^{-(1+\kappa/8)+o(1)}$$ for $$\kappa\in[0,8)$$ along some sequence $$\varepsilon\searrow 0$$. The paper concludes with a set of interesting conjectures.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics 60J60 Diffusion processes
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