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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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