施拉姆-勒夫纳演进

在概率论中，施拉姆-勒夫纳演变（Schramm–Loewner evolution，SLE）是一个平面曲线的家族以及统计力学模型的缩放极限。

应用

Uniform spanning tree, Loop erased random walk（英语：Loop erased random walk）
自避行走
普遍性 (物理学)
施拉姆-勒夫纳进化描述临界渗流，临界易辛模型，自避行走的缩放极限
统计力学模型
因为SLE有马尔可夫性质，所以可以用伊藤微积分来分析一下
共形场论

勒夫纳演变

D 是单连通的开集。D是复杂域，但是不等于C。
γ 是D中的一条曲线。γ 在D 的边界开始。
$D_{t}=D-\gamma ([0,t])$
因为 $D_{t}$ 是单连通的，它通过共形映射等于D（黎曼映射理论）。
$f:D_{t}\to D$ 是同构。
$g(f(z))=z$ 是反函数。
在t = 0，f₀(z) = z 和 g₀(z) = z。
ζ(t)是驱动函数（driving function），接受D边界上的值。

根据Loewner (1923，p. 121)，Loewner方程（英语：Loewner differential equation）是

{\frac {\partial f_{t}(z)}{\partial t}}=-zf_{t}^{\prime }(z){\frac {\zeta (t)+z}{\zeta (t)-z}}

{\dfrac {\partial g_{t}(z)}{\partial t}}=g_{t}(z){\dfrac {\zeta (t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.

$\zeta ,\gamma$ 的关系是

f_{t}(\zeta (t))=\gamma (t){\text{ 或  }}\ \zeta (t)=g_{t}(\gamma (t))

施拉姆-勒夫纳演变

SL演变是一个勒夫纳方程，有下面的驱动函数

$\zeta (t)={\sqrt {\kappa }}B(t)$

其中 B(t) 是D边界上的布朗运动。

例如

若0 ≤ κ ≤ 4，曲线γ(t)几乎必然是简单曲线
若4 < κ < 8，γ(t) 与自身相交。
若 κ ≥ 8，γ(t)是space-filling的。
若κ = 2，曲线是Loop-erased random walk。^[1]^[2]
κ = 8：皮亚诺曲线
若 κ = 8/3，有人猜想这个SLE描述自避行走。
κ = 3：易辛模型边界的极限
κ = 4：高斯自由场，harmonic explorer (2005),^[3]
κ = 6：斯坦尼斯拉·斯米尔诺夫证明SLE₆ 是格子（正三角形镶嵌）上的临界渗透的缩放极限^[4]^[5]，计算临界指数^[6]^[7]^[8]；证明渗流的共形不变性Smirnov (2001)^[9]，Cardy方程
κ = 8：path separating UST from dual tree

属性

若SLE描述共形场论，central charge c等于

c={\frac {(8-3\kappa )(\kappa -6)}{2\kappa }}.

Beffara (2008) 表明了SLE的豪斯多夫维数是min(2, 1 + κ/8)。

Lawler，Schramm & Werner (2001) 用SLE₆ 证明Mandelbrot (1982)的猜想：平面布朗运动边界的分形维数是4/3。

Rohde和Schramm表明了曲线的分形维数是

d=1+{\frac {\kappa }{8}}.

模拟

https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution（页面存档备份，存于互联网档案馆）

参考文献

^ Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 2004, 32 (1B): 939–995. arXiv:math/0112234  . doi:10.1214/aop/1079021469.
^ Kenyon, Richard. Long range properties of spanning trees. J. Math. Phys. 2000, 41 (3): 1338–1363. Bibcode:10.1.1.39.7560 请检查|bibcode=值 (帮助). doi:10.1063/1.533190.
^ Schramm, Oded; Sheffield, Scott, Harmonic explorer and its convergence to SLE4., Annals of Probability, 2005, 33 (6): 2127–2148, JSTOR 3481779, arXiv:math/0310210  , doi:10.1214/009117905000000477
^ Smirnov, Stanislav. Critical percolation in the plane. Comptes Rendus de l'Académie des Sciences. 2001, 333 (3): 239–244. Bibcode:2001CRASM.333..239S. arXiv:0909.4499  . doi:10.1016/S0764-4442(01)01991-7.
^ Kesten, Harry. Scaling relations for 2D-percolation. Comm. Math. Phys. 1987, 109 (1): 109–156. Bibcode:1987CMaPh.109..109K. doi:10.1007/BF01205674.
^ Smirnov, Stanislav; Werner, Wendelin. Critical exponents for two-dimensional percolation. Math. Res. Lett. 2001, 8 (6): 729–744 [2020-02-11]. arXiv:math/0109120  . doi:10.4310/mrl.2001.v8.n6.a4. （原始内容 (PDF)存档于2021-03-08）.
^ Schramm, Oded; Steif, Jeffrey E. Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. 2010, 171 (2): 619–672. arXiv:math/0504586  . doi:10.4007/annals.2010.171.619.
^ Garban, Christophe; Pete, Gábor; Schramm, Oded. Pivotal, cluster and interface measures for critical planar percolation. J. Amer. Math. Soc. 2013, 26 (4): 939–1024. arXiv:1008.1378  . doi:10.1090/S0894-0347-2013-00772-9.
^ Smirnov, Stanislav. Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. Comptes Rendus de l'Académie des Sciences, Série I. 2001, 333 (3): 239–244. Bibcode:2001CRASM.333..239S. ISSN 0764-4442. arXiv:0909.4499  . doi:10.1016/S0764-4442(01)01991-7.

阅读

https://terrytao.wordpress.com/tag/schramm-loewner-evolution/（页面存档备份，存于互联网档案馆）（页面存档备份，存于互联网档案馆）（陶哲轩介绍SLE）
http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf（页面存档备份，存于互联网档案馆）（页面存档备份，存于互联网档案馆）（Conformally invariant process in plane, by Lawler）
http://pi.math.cornell.edu/~cpss/2011/lawler-notes.pdf（SCALING LIMITS AND THE SCHRAMM-LOEWNER EVOLUTION GREGORY F. LAWLER）

Beffara, Vincent, The dimension of the SLE curves, The Annals of Probability, 2008, 36 (4): 1421–1452, MR 2435854, arXiv:math/0211322  , doi:10.1214/07-AOP364
Cardy, John, SLE for theoretical physicists, Annals of Physics, 2005, 318 (1): 81–118, Bibcode:2005AnPhy.318...81C, arXiv:cond-mat/0503313  , doi:10.1016/j.aop.2005.04.001
Hazewinkel, Michiel (编), 施拉姆-勒夫纳演进, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
Hazewinkel, Michiel (编), 施拉姆-勒夫纳演进, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
Kager, Wouter; Nienhuis, Bernard, A Guide to Stochastic Loewner Evolution and its Applications, J. Stat. Phys., 2004, 115 (5/6): 1149–1229, Bibcode:2004JSP...115.1149K, arXiv:math-ph/0312056  , doi:10.1023/B:JOSS.0000028058.87266.be
Lawler, Gregory F., An introduction to the stochastic Loewner evolution, Kaimanovich, Vadim A. (编), Random walks and geometry, Walter de Gruyter GmbH & Co. KG, Berlin: 261–293, 2004 [2020-02-11], ISBN 978-3-11-017237-9, MR 2087784, （原始内容存档于2009-09-18）
Lawler, Gregory F., Conformally invariant processes in the plane, Mathematical Surveys and Monographs 114, Providence, R.I.: American Mathematical Society, 2005, ISBN 978-0-8218-3677-4, MR 2129588
Lawler, Gregory F., Schramm–Loewner Evolution, 2007, arXiv:0712.3256  [math.PR]
Lawler, Gregory F., Stochastic Loewner Evolution, [2020-02-11], （原始内容存档于2016-03-04）
Lawler, Gregory F., Conformal invariance and 2D statistical physics, Bull. Amer. Math. Soc., 2009, 46: 35–54, doi:10.1090/S0273-0979-08-01229-9
Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin, The dimension of the planar Brownian frontier is 4/3, Mathematical Research Letters, 2001, 8 (4): 401–411 [2020-02-11], MR 1849257, arXiv:math/0010165  , doi:10.4310/mrl.2001.v8.n4.a1, （原始内容存档于2019-09-08）
Loewner, C., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I (PDF), Math. Ann., 1923, 89 (1–2): 103–121 [2020-02-11], JFM 49.0714.01, doi:10.1007/BF01448091, （原始内容存档 (PDF)于2019-09-26）
Mandelbrot, Benoît, The Fractal Geometry of Nature, W. H. Freeman, 1982, ISBN 978-0-7167-1186-5
Norris, J. R., Introduction to Schramm–Loewner evolutions (PDF), 2010 [2020-02-11], （原始内容存档 (PDF)于2019-07-14）
Pommerenke, Christian, Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher 15, Vandenhoeck & Ruprecht, 1975 (Chapter 6 treats the classical theory of Loewner's equation)
Schramm, Oded, Scaling limits of loop-erased random walks and uniform spanning trees, Israel Journal of Mathematics, 2000, 118: 221–288, MR 1776084, arXiv:math.PR/9904022  , doi:10.1007/BF02803524 Schramm's original paper, introducing SLE
Schramm, Oded, Conformally invariant scaling limits: an overview and a collection of problems, International Congress of Mathematicians. Vol. I, Eur. Math. Soc., Zürich: 513–543, 2007, ISBN 978-3-03719-022-7, MR 2334202, arXiv:math/0602151  , doi:10.4171/022-1/20
Werner, Wendelin, Random planar curves and Schramm–Loewner evolutions, Lectures on probability theory and statistics, Lecture Notes in Math. 1840, Berlin, New York: Springer-Verlag: 107–195, 2004, ISBN 978-3-540-21316-1, MR 2079672, arXiv:math.PR/0303354  , doi:10.1007/b96719
Werner, Wendelin, Conformal restriction and related questions, Probability Surveys, 2005, 2: 145–190, MR 2178043, doi:10.1214/154957805100000113

[LERW-1] Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 2004, 32 (1B): 939–995. arXiv:math/0112234  . doi:10.1214/aop/1079021469.

[2] Kenyon, Richard. Long range properties of spanning trees. J. Math. Phys. 2000, 41 (3): 1338–1363. Bibcode:10.1.1.39.7560 请检查|bibcode=值 (帮助). doi:10.1063/1.533190.

[Harmonic_explorer_2005-3] Schramm, Oded; Sheffield, Scott, Harmonic explorer and its convergence to SLE4., Annals of Probability, 2005, 33 (6): 2127–2148, JSTOR 3481779, arXiv:math/0310210  , doi:10.1214/009117905000000477

[4] Smirnov, Stanislav. Critical percolation in the plane. Comptes Rendus de l'Académie des Sciences. 2001, 333 (3): 239–244. Bibcode:2001CRASM.333..239S. arXiv:0909.4499  . doi:10.1016/S0764-4442(01)01991-7.

[5] Kesten, Harry. Scaling relations for 2D-percolation. Comm. Math. Phys. 1987, 109 (1): 109–156. Bibcode:1987CMaPh.109..109K. doi:10.1007/BF01205674.

[6] Smirnov, Stanislav; Werner, Wendelin. Critical exponents for two-dimensional percolation. Math. Res. Lett. 2001, 8 (6): 729–744 [2020-02-11]. arXiv:math/0109120  . doi:10.4310/mrl.2001.v8.n6.a4. （原始内容 (PDF)存档于2021-03-08）.

[7] Schramm, Oded; Steif, Jeffrey E. Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. 2010, 171 (2): 619–672. arXiv:math/0504586  . doi:10.4007/annals.2010.171.619.

[8] Garban, Christophe; Pete, Gábor; Schramm, Oded. Pivotal, cluster and interface measures for critical planar percolation. J. Amer. Math. Soc. 2013, 26 (4): 939–1024. arXiv:1008.1378  . doi:10.1090/S0894-0347-2013-00772-9.

[Smirnov2001-9] Smirnov, Stanislav. Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. Comptes Rendus de l'Académie des Sciences, Série I. 2001, 333 (3): 239–244. Bibcode:2001CRASM.333..239S. ISSN 0764-4442. arXiv:0909.4499  . doi:10.1016/S0764-4442(01)01991-7.

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