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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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