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The dimension of the SLE curves. (English) Zbl 1165.60007
Let the random curve $$\gamma:[0,\infty)\to \overline{\mathbb{H}}$$ be a trace of the chordal $$SLE_{\kappa}$$ where $$\overline{\mathbb{H}}$$ is a closure of the upper-half plane $$\mathbb{H}$$ (for details concerning the Schramm–Loewner Evolution, i.e. a family of maps $$\{g_t,t\geq 0\}$$ called $$SLE_{\kappa}$$ and introduced by the following PDE: $\partial_tg_t(z) = \frac{2}{g_t(z) - \sqrt{\kappa} B_t},\;\;g_0(z)=z,$ where $$(B_t)$$ is a standard Brownian motion and $$\kappa$$ is a positive parameter, see, e.g., [S. Rohde, O. Schramm, Ann. Math, (2) 161, 883–924 (2005; Zbl 1081.60069)]. In the above mentioned paper the authors proved that the Hausdorff dimension of the $$SLE_{\kappa}$$ trace is not larger than $$1+\frac{\kappa}{8}$$ when $$\kappa \leq 8$$, and they conjectured the sharpness of this bound. The main result of the present paper provides the proof of this conjecture. Namely, almost surely one has $$\dim_H(\mathcal{H}) = 2 \wedge (1+ \frac{\kappa}{8})$$ where $$\mathcal{H}= \gamma([0,\infty))$$. This result was known for $$\kappa \geq 8$$ because the curve is then space-filling.

##### MSC:
 60D05 Geometric probability and stochastic geometry 60G17 Sample path properties 28A80 Fractals
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