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。 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 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 − γ ( [ 0 , t ] ) {\displaystyle D_{t}=D-\gamma ([0,t])} 因为 D t {\displaystyle 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 → D {\displaystyle f:D_{t}\to D} 是同构。 g ( f ( z ) ) = z {\displaystyle g(f(z))=z} 是反函數。 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 ∂ f t ( z ) ∂ t = − z f t ′ ( z ) ζ ( t ) + z ζ ( t ) − z {\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}=-zf_{t}^{\prime }(z){\frac {\zeta (t)+z}{\zeta (t)-z}}} ∂ g t ( z ) ∂ t = g t ( z ) ζ ( t ) + g t ( z ) ζ ( t ) − g t ( z ) . {\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}=g_{t}(z){\dfrac {\zeta (t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.} ζ , γ {\displaystyle \zeta ,\gamma } 的关系是 f t ( ζ ( t ) ) = γ ( t )  或    ζ ( t ) = g t ( γ ( t ) ) {\displaystyle f_{t}(\zeta (t))=\gamma (t){\text{ 或 }}\ \zeta (t)=g_{t}(\gamma (t))} SL演变是一个勒夫纳方程，有下面的驱动函数 ζ ( t ) = κ B ( t ) {\displaystyle \zeta (t)={\sqrt {\kappa }}B(t)} 其中 B(t) 是D边界上的布朗运动。 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。 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) harvtxt error: no target: CITEREFharmonic_explorer2005 (help), κ = 6：斯坦尼斯拉·斯米尔诺夫证明SLE6 是格子（正三角形鑲嵌）上的临界渗透的缩放极限，计算临界指数；证明渗流的共形不变性Smirnov (2001) harvtxt error: multiple targets (2×): CITEREFSmirnov2001 (help)，Cardy方程 κ = 8：path separating UST from dual tree 若SLE描述共形场论，central charge c等于 c = ( 8 − 3 κ ) ( κ − 6 ) 2 κ . {\displaystyle 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 + κ 8 . {\displaystyle d=1+{\frac {\kappa }{8}}.} https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution（页面存档备份，存于互联网档案馆） 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:2000JMP....41.1338K. CiteSeerX 10.1.1.39.7560 . 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.  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[失效連結]（SCALING LIMITS AND THE SCHRAMM-LOEWNER EVOLUTION GREGORY F. 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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. 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