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摘要翻译：
标准的Schramm-Loewner演化(SLE)是由一个连续的布朗运动驱动的，布朗运动产生一个轨迹，一个连接运动奇异点的连续分形曲线。如果将跳转添加到驱动函数中，则跟踪分支。在最近的出版物[1]中，我们引入了一个由布朗运动和分形跳跃集叠加驱动的广义SLE（从技术上讲是一个稳定的L\'evy过程）。然后我们讨论了由此产生的L\'Evy-SLE生长过程的小尺度性质。这里我们讨论的是同样的模型，但重点是随着时间的无限大而发生的全局伸缩行为。这种限制行为与布朗强迫无关，只依赖于定义稳定L\'evy分布形状的单个参数$\\alpha$。我们通过研究一个Fokker-Planck方程来了解这种行为，该方程给出了轨迹端点随时间变化的概率分布。与先前研究过的短时情形一样，我们观察到这个生长过程的性质在$\alpha=1$时发生了质的和奇异的变化。我们从分析和数值上说明，对于$\alpha>1$，增长在垂直方向上无限地持续，对于$\alpha=1$，增长为$\log t$，对于$\alpha<1$，增长为饱和。概率密度有两个不同的尺度，分别对应于沿边界和垂直边界的方向。在前一种情况下，特征刻度为$x(t)\sim t^{1/\alpha}$。在后一种情况下，小数位数为$y(t)\sim A+B t^{1-1/\alpha}$（对于$\alpha\neq1$)和$y(t)\sim\ln t$（对于$\alpha=1$）。给出了各种极限情况下概率密度的标度函数。
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英文标题：
《Global properties of Stochastic Loewner evolution driven by Levy
processes》
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作者：
P. Oikonomou, I. Rushkin, I. A. Gruzberg, L. P. Kadanoff
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最新提交年份：
2008
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分类信息：
一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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英文摘要：
Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication [1] we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable L\'evy process). We then discussed the small-scale properties of the resulting L\'evy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, $\alpha$, which defines the shape of the stable L\'evy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for endpoints of the trace as a function of time. As in the short-time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at $\alpha =1$. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for $\alpha > 1$, goes as $\log t$ for $\alpha = 1$, and saturates for $\alpha< 1$. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is $X(t) \sim t^{1/\alpha}$. In the latter case the scale is $Y(t) \sim A + B t^{1-1/\alpha}$ for $\alpha \neq 1$, and $Y(t) \sim \ln t$ for $\alpha = 1$. Scaling functions for the probability density are given for various limiting cases.
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PDF链接：
https://arxiv.org/pdf/710.268
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