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
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。
- 因为 是单连通的,它通过共形映射等于D(黎曼映射理论)。
- 是同构。
- 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方程是
的关系是
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
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(正三角形鑲嵌)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
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
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: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.
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[失效連結](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