施拉姆-勒夫納演進
在概率論中,施拉姆-勒夫納演變(Schramm–Loewner evolution,SLE)是一個平面曲線的家族以及統計力學模型的縮放極限。
應用
- Uniform spanning tree, Loop erased random walk
- 自避行走
- 普遍性 (物理學)
- 施拉姆-勒夫納進化描述臨界滲流,臨界易辛模型,自避行走的縮放極限
- 統計力學模型
- 因為SLE有馬爾可夫性質,所以可以用伊藤微積分來分析一下
- 共形場論
勒夫納演變
- D 是單連通的開集。D是複雜域,但是不等於C。
- γ 是D中的一條曲線。γ 在D 的邊界開始。
- 因為是單連通的,它通過共形映射等於D(黎曼映射理論)。
- 是同構。
- 是反函數。
- 在t = 0,f0(z) = z 和 g0(z) = z。
- ζ(t)是驅動函數(driving function),接受D邊界上的值。
根據Loewner (1923,p. 121),Loewner方程是
的關係是
施拉姆-勒夫納演變
SL演變是一個勒夫納方程,有下面的驅動函數
其中 B(t) 是D邊界上的布朗運動。
例如
- 若0 ≤ κ ≤ 4,曲線γ(t)幾乎必然是簡單曲線
- 若4 < κ < 8,γ(t) 與自身相交。
- 若 κ ≥ 8,γ(t)是space-filling的。
- 若κ = 2,曲線是Loop-erased random walk。[1][2]
- κ = 8:皮亞諾曲線
- 若 κ = 8/3,有人猜想這個SLE描述自避行走。
- κ = 3:易辛模型邊界的極限
- κ = 4:高斯自由場,harmonic explorer (2005),[3]
- κ = 6:斯坦尼斯拉·斯米爾諾夫證明SLE6 是格子(正三角形鑲嵌)上的臨界滲透的縮放極限[4][5],計算臨界指數[6][7][8];證明滲流的共形不變性Smirnov (2001)[9],Cardy方程
- κ = 8:path separating UST from dual tree
屬性
若SLE描述共形場論,central charge c等於
Beffara (2008) 表明了SLE的豪斯多夫維數是min(2, 1 + κ/8)。
Lawler,Schramm & Werner (2001) 用SLE6 證明Mandelbrot (1982)的猜想:平面布朗運動邊界的分形維數是4/3。
Rohde和Schramm表明了曲線的分形維數是
模擬
https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution(頁面存檔備份,存於互聯網檔案館)
參考文獻
- ^ 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. (原始內容 (PDF)存檔於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.
閱讀
- https://terrytao.wordpress.com/tag/schramm-loewner-evolution/(頁面存檔備份,存於互聯網檔案館) (頁面存檔備份,存於互聯網檔案館)(陶哲軒介紹SLE)
- http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf(頁面存檔備份,存於互聯網檔案館) (頁面存檔備份,存於互聯網檔案館)(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 (編), 施拉姆-勒夫纳演进, 数学百科全书, Springer, 2001, ISBN 978-1-55608-010-4
- Hazewinkel, Michiel (編), 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. (編), 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, (原始內容存檔於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], (原始內容存檔於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, (原始內容存檔於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, (原始內容存檔 (PDF)於2019-09-26)
- Mandelbrot, Benoît, The Fractal Geometry of Nature, W. H. Freeman, 1982, ISBN 978-0-7167-1186-5
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- 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
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, doi:10.1007/b96719
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