论文标题
连续证明SLE $ _8 $曲线的存在
A continuous proof of the existence of the SLE$_8$ curve
论文作者
论文摘要
假设$η$是一个全平面空间填充SLE $_κ$，对于$κ\ in（4,8）$，从$ \ infty $到$ \ fty $ \ \ \ \ \ infty $由lebesgue Measure量化并归一化，以便$η（0）= 0 $。对于（4,8）$中的每个$ t> 0 $和$κ\，我们让$μ__{κ，t} $表示$η| _ {[0，t]} $的定律。我们向每个$ν，t> 0 $显示，法律家族$μ_{κ，t} $ for $κ\ in [4+ν，8）$在与连续曲线上的概率度量$ [0，t] \ to {\ Mathbf C} $配备的较弱范围相关的弱拓扑中是紧凑的。作为这种紧密度结果的直接副产品（以$κ\ uparrow 8 $的限制），我们获得了SLE $ _8 $曲线存在的新证明，该证明不基于Lawler-Schramm-Werner的离散统一树的缩放限制。
Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $κ\uparrow 8$), we obtain a new proof of the existence of the SLE$_8$ curve which does not build on the discrete uniform spanning tree scaling limit of Lawler-Schramm-Werner.
