$\emph{\textbf{SLE}}_{\bm {\kappa}}$\,与临界渗流的右路径概率

华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (3): 24-29.

$\emph{\textbf{SLE}}_{\bm {\kappa}}$\,与临界渗流的右路径概率

梁静$^1,2, 蓝师义²

1. 淮南师范学院~~数学与计算机科学系, 安徽~~淮南 232001; 2. 广西民族大学~~数学与计算机科学学院, 南宁 530006

收稿日期:2011-05-01 修回日期:2011-08-01 出版日期:2012-05-25 发布日期:2012-05-22

Right-passage probabilities of $\emph{\textbf{SLE}}_{\bm {\kappa}}$ and critical percolation

LIANG Jing ^1,2, LAN Shi-yi ²

1. Department of Mathematical and Computer Sciencs, Huainan Normal University, Huainan Anhui} 232001, China; 2. School of Mathematical and Computer Science, Guangxi University for Nationalities, Nanning 530006, China

Received:2011-05-01 Revised:2011-08-01 Online:2012-05-25 Published:2012-05-22

摘要/Abstract

摘要： 给定上半平面内的一个固定点, 获得通弦~$SLE_{\kappa}(0\leqslant \kappa<8)$~迹穿过它右边的概率估计公式. 基于左边界概率的结果, 建立了闭单位圆内临界渗流不包含其内一个固定点的概率估计公式. 最后, 利用探索过程与~$SLE_{6}$~的关系, 得到了起点和终点相同的~$SLE_{6}$~迹与自避型路径有同样的分布.

关键词: 通弦~$SLE_\kappa$, 洛纳方程, 调和测度, 临界渗流

Abstract: This paper derived the probability formula for a chordal $SLE_{\kappa}$ trace across a given point in upper half plane $\mathbb{H}$. And on the basis of the left-passage probability, established the probability formula for a critical percolation in a closed unit circle without a given point. Finally, according to the relationship of exploration process and $SLE_{6}$, got that with the same starting and ending point, the trace of $SLE_{6}$ has the same distribution of self-avoiding walk.

Key words: chrodal $SLE_{\kappa}$, Loewner equation, harmonic measure, critical percolation

中图分类号:

梁静, 蓝师义. $\emph{\textbf{SLE}}_{\bm {\kappa}}$\,与临界渗流的右路径概率[J]. 华东师范大学学报(自然科学版), 2012, 2012(3): 24-29.