We study $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$ curves, with $$\kappa $$ and $$\rho $$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed “angle” and determine the almost sure Hausdorff dimensions of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $$g_t$$, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

中文翻译：

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$${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$ 的多重分形边界谱

我们研究 $${{\,\mathrm{SLE}\,}}_\kappa (\rho )$$ 曲线，选择 $$\kappa $$ 和 $$\rho $$ 使曲线达到边界. 更准确地说，我们研究曲线以规定“角度”与边界碰撞的集合，并确定这些集合的几乎确定的 Hausdorff 维数。这是通过研究保形图 $$g_t$$ 的空间导数的矩来完成的，通过采用 Girsanov 定理并使用虚几何技术来推导相关性估计。