施拉姆-勒夫纳演进:修订间差异
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== 应用 == |
== 应用 == |
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*[[自避行走]] |
*[[自避行走]] |
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*[[普遍性 (物理学)]] |
*[[普遍性 (物理学)]] |
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* |
* 施拉姆-勒夫纳进化描述[[渗流理论|临界渗流]],临界[[易辛模型]],[[自避行走]]的[[缩放极限]] |
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* [[统计力学]]模型 |
* [[统计力学]]模型 |
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* 因为SLE有[[马尔可夫性质]],所以可以用[[伊藤积分|伊藤微积分]]来分析一下 |
* 因为SLE有[[马尔可夫性质]],所以可以用[[伊藤积分|伊藤微积分]]来分析一下 |
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* [[共形场论]] |
* [[共形场论]] |
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== |
== 勒夫纳演变 == |
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* ''D'' 是[[單連通|单连通]]的[[开集]]。D是[[复数 (数学)|复杂域]],但是不等于'''C。''' |
* ''D'' 是[[單連通|单连通]]的[[开集]]。D是[[复数 (数学)|复杂域]],但是不等于'''C。''' |
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:<math> f_t(\zeta(t)) = \gamma(t) \text{ 或 }\ \zeta(t) = g_t(\gamma(t)) </math> |
:<math> f_t(\zeta(t)) = \gamma(t) \text{ 或 }\ \zeta(t) = g_t(\gamma(t)) </math> |
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== |
== 施拉姆-勒夫纳演变 == |
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SL演变是一个 |
SL演变是一个勒夫纳方程,有下面的驱动函数 |
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<math>\zeta(t)=\sqrt{\kappa}B(t)</math> |
<math>\zeta(t)=\sqrt{\kappa}B(t)</math> |
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* 若4 < ''κ'' < 8,γ(''t'') 与自身相交。 |
* 若4 < ''κ'' < 8,γ(''t'') 与自身相交。 |
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* 若 ''κ'' ≥ 8,γ(''t'')是space-filling的。 |
* 若 ''κ'' ≥ 8,γ(''t'')是space-filling的。 |
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* 若''κ'' = 2,曲线是Loop-erased random walk。<ref name="LERW">{{Cite journal|title=Conformal invariance of planar loop-erased random walks and uniform spanning trees|last=Lawler|first=Gregory F.|last2=Schramm|first2=Oded|journal=[[Annals of Probability|Ann. Probab.]]|issue=1B|doi=10.1214/aop/1079021469|year=2004|volume=32|pages=939–995|arxiv=math/0112234|last3=Werner|first3=Wendelin}}</ref><ref>{{ |
* 若''κ'' = 2,曲线是Loop-erased random walk。<ref name="LERW">{{Cite journal|title=Conformal invariance of planar loop-erased random walks and uniform spanning trees|last=Lawler|first=Gregory F.|last2=Schramm|first2=Oded|journal=[[Annals of Probability|Ann. Probab.]]|issue=1B|doi=10.1214/aop/1079021469|year=2004|volume=32|pages=939–995|arxiv=math/0112234|last3=Werner|first3=Wendelin}}</ref><ref>{{cite journal |first=Richard |last=Kenyon |title=Long range properties of spanning trees |journal=[[Journal of Mathematical Physics|J. Math. Phys.]] |volume=41 |issue=3 |pages=1338–1363 |year=2000 |doi=10.1063/1.533190 |bibcode=2000JMP....41.1338K |citeseerx=10.1.1.39.7560 }}</ref> |
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* ''κ'' = 8:[[皮亚诺曲线]] |
* ''κ'' = 8:[[皮亚诺曲线]] |
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* 若 ''κ'' = 8/3,有人猜想这个SLE描述[[自避行走]]。 |
* 若 ''κ'' = 8/3,有人猜想这个SLE描述[[自避行走]]。 |
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* ''κ'' = 3:[[易辛模型]]边界的极限 |
* ''κ'' = 3:[[易辛模型]]边界的极限 |
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* ''κ'' = 4:[[高斯自由场]],{{Harvard citation text|harmonic explorer|2005}},<ref name="Harmonic explorer 2005">{{Citation|last=Schramm|first=Oded|last2=Sheffield|first2=Scott|title=Harmonic explorer and its convergence to SLE4.|jstor=3481779|year=2005|journal=Annals of Probability|volume=33|number=6|pages=2127–2148|doi=10.1214/009117905000000477|arxiv=math/0310210}}</ref> |
* ''κ'' = 4:[[高斯自由场]],{{Harvard citation text|harmonic explorer|2005}},<ref name="Harmonic explorer 2005">{{Citation|last=Schramm|first=Oded|last2=Sheffield|first2=Scott|title=Harmonic explorer and its convergence to SLE4.|jstor=3481779|year=2005|journal=Annals of Probability|volume=33|number=6|pages=2127–2148|doi=10.1214/009117905000000477|arxiv=math/0310210}}</ref> |
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* ''κ'' = 6:[[斯坦尼斯拉夫·斯米尔诺夫|斯坦尼斯拉·斯米尔诺夫]]证明SLE<sub>6</sub> 是[[格子]]([[正三角形鑲嵌]])上的[[渗流理论|临界渗透]]的[[缩放极限]]<ref>{{Cite journal|title=Critical percolation in the plane|last=Smirnov|first=Stanislav|journal=Comptes Rendus de l'Académie des Sciences|issue=3|doi=10.1016/S0764-4442(01)01991-7|year=2001|volume=333|pages=239–244|arxiv=0909.4499|bibcode=2001CRASM.333..239S}}</ref><ref>{{Cite journal|title=Scaling relations for 2D-percolation|last=Kesten|first=Harry|journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]]|issue=1|doi=10.1007/BF01205674|year=1987|volume=109|pages=109–156|bibcode=1987CMaPh.109..109K}}</ref>,计算[[临界指数]]<ref>{{Cite journal|title=Critical exponents for two-dimensional percolation|url=http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2001/0008/0006/MRL-2001-0008-0006-00019853.pdf|last=Smirnov|first=Stanislav|last2=Werner|first2=Wendelin|journal=[[Mathematical Research Letters|Math. Res. Lett.]]|issue=6|doi=10.4310/mrl.2001.v8.n6.a4|year=2001|volume=8|pages=729–744|arxiv=math/0109120 |
* ''κ'' = 6:[[斯坦尼斯拉夫·斯米尔诺夫|斯坦尼斯拉·斯米尔诺夫]]证明SLE<sub>6</sub> 是[[格子]]([[正三角形鑲嵌]])上的[[渗流理论|临界渗透]]的[[缩放极限]]<ref>{{Cite journal|title=Critical percolation in the plane|last=Smirnov|first=Stanislav|journal=Comptes Rendus de l'Académie des Sciences|issue=3|doi=10.1016/S0764-4442(01)01991-7|year=2001|volume=333|pages=239–244|arxiv=0909.4499|bibcode=2001CRASM.333..239S}}</ref><ref>{{Cite journal|title=Scaling relations for 2D-percolation|url=https://archive.org/details/sim_communications-in-mathematical-physics_1987-03_109_1/page/109|last=Kesten|first=Harry|journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]]|issue=1|doi=10.1007/BF01205674|year=1987|volume=109|pages=109–156|bibcode=1987CMaPh.109..109K}}</ref>,计算[[临界指数]]<ref>{{Cite journal|title=Critical exponents for two-dimensional percolation|url=http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2001/0008/0006/MRL-2001-0008-0006-00019853.pdf|last=Smirnov|first=Stanislav|last2=Werner|first2=Wendelin|journal=[[Mathematical Research Letters|Math. Res. Lett.]]|issue=6|doi=10.4310/mrl.2001.v8.n6.a4|year=2001|volume=8|pages=729–744|arxiv=math/0109120|access-date=2020-02-11|archive-date=2021-03-08|archive-url=https://web.archive.org/web/20210308065326/https://arxiv.org/abs/math/0109120|dead-url=yes}}</ref><ref>{{Cite journal|title=Quantitative noise sensitivity and exceptional times for percolation|url=https://archive.org/details/sim_annals-of-mathematics_2010-03_171_2/page/619|last=Schramm|first=Oded|last2=Steif|first2=Jeffrey E.|journal=[[Annals of Mathematics|Ann. of Math.]]|issue=2|doi=10.4007/annals.2010.171.619|year=2010|volume=171|pages=619–672|arxiv=math/0504586}}</ref><ref>{{Cite journal|title=Pivotal, cluster and interface measures for critical planar percolation|last=Garban|first=Christophe|last2=Pete|first2=Gábor|journal=[[Journal of the American Mathematical Society|J. Amer. Math. Soc.]]|issue=4|doi=10.1090/S0894-0347-2013-00772-9|year=2013|volume=26|pages=939–1024|arxiv=1008.1378|last3=Schramm|first3=Oded}}</ref>;证明[[渗流]]的共形不变性{{Harvard citation text|Smirnov|2001}}<ref name="Smirnov2001">{{Cite journal|title=Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits|last=Smirnov|first=Stanislav|journal=Comptes Rendus de l'Académie des Sciences, Série I|issue=3|doi=10.1016/S0764-4442(01)01991-7|year=2001|volume=333|pages=239–244|arxiv=0909.4499|bibcode=2001CRASM.333..239S|issn=0764-4442}}</ref>,Cardy方程 |
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* ''κ'' = 8:path separating UST from dual tree |
* ''κ'' = 8:path separating UST from dual tree |
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== 模拟 == |
== 模拟 == |
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https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution |
https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution{{Wayback|url=https://github.com/xsources/Matlab-simulation-of-Schramm-Loewner-Evolution |date=20200914025701 }} |
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== 参考文献 == |
== 参考文献 == |
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== 阅读 == |
== 阅读 == |
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* https://terrytao.wordpress.com/tag/schramm-loewner-evolution/<nowiki/>([[陶哲轩]]介绍SLE) |
* https://terrytao.wordpress.com/tag/schramm-loewner-evolution/{{Wayback|url=https://terrytao.wordpress.com/tag/schramm-loewner-evolution/ |date=20190202182332 }} {{Wayback|url=https://terrytao.wordpress.com/tag/schramm-loewner-evolution/ |date=20190202182332 }}<nowiki/>([[陶哲轩]]介绍SLE) |
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* http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf<nowiki/>(Conformally invariant process in plane, by Lawler) |
* http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf{{Wayback|url=http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf |date=20180304215642 }} {{Wayback|url=http://users.ictp.it/~pub_off/lectures/lns017/Lawler/Lawler.pdf |date=20180304215642 }}<nowiki/>(Conformally invariant process in plane, by Lawler) |
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* http://pi.math.cornell.edu/~cpss/2011/lawler-notes.pdf<nowiki/>(SCALING LIMITS AND THE SCHRAMM-LOEWNER EVOLUTION GREGORY F. LAWLER) |
* http://pi.math.cornell.edu/~cpss/2011/lawler-notes.pdf{{Dead link}}<nowiki/>(SCALING LIMITS AND THE SCHRAMM-LOEWNER EVOLUTION GREGORY F. LAWLER) |
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{{ReflistH}} |
{{ReflistH}} |
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*{{Citation|last=Beffara|first=Vincent|title=The dimension of the SLE curves|mr=2435854|year=2008|journal=The Annals of Probability|volume=36|number=4|pages=1421–1452|doi=10.1214/07-AOP364|arxiv=math/0211322}} |
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*{{Springer}} |
*{{Springer}} |
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*{{Citation|arxiv=math-ph/0312056|title=A Guide to Stochastic Loewner Evolution and its Applications|first=Wouter|last=Kager|first2=Bernard|last2=Nienhuis|journal=J. Stat. Phys.|volume=115|pages=1149–1229|year=2004|doi=10.1023/B:JOSS.0000028058.87266.be|number=5/6|bibcode=2004JSP...115.1149K}} |
*{{Citation|arxiv=math-ph/0312056|title=A Guide to Stochastic Loewner Evolution and its Applications|first=Wouter|last=Kager|first2=Bernard|last2=Nienhuis|journal=J. Stat. Phys.|volume=115|pages=1149–1229|year=2004|doi=10.1023/B:JOSS.0000028058.87266.be|number=5/6|bibcode=2004JSP...115.1149K}} |
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*{{Citation|last=Lawler|first=Gregory F.|title=Random walks and geometry|chapter-url=http://www.math.duke.edu/~jose/esi.html|publisher=Walter de Gruyter GmbH & Co. KG, Berlin|mr=2087784|year=2004|chapter=An introduction to the stochastic Loewner evolution|pages=261–293|editor1-first=Vadim A.|editor1-last=Kaimanovich|isbn=978-3-11-017237-9|archiveurl=https://web.archive.org/web/20090918041127/http://www.math.duke.edu/~jose/esi.html|archivedate=2009-09-18|accessdate=2020-02-11|dead-url=no}} |
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*{{Citation|last=Lawler|first=Gregory F.|title=Conformally invariant processes in the plane|publisher=[[American Mathematical Society]]|place=Providence, R.I.|series=Mathematical Surveys and Monographs|isbn=978-0-8218-3677-4|mr=2129588|year=2005|volume=114|url=https://books.google.com/books?id=JHMzab3u6U8C}} |
*{{Citation|last=Lawler|first=Gregory F.|title=Conformally invariant processes in the plane|publisher=[[American Mathematical Society]]|place=Providence, R.I.|series=Mathematical Surveys and Monographs|isbn=978-0-8218-3677-4|mr=2129588|year=2005|volume=114|url=https://books.google.com/books?id=JHMzab3u6U8C}} |
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*{{Cite arXiv |last1=Lawler |first1=Gregory F. |title=Schramm–Loewner Evolution |year=2007 |eprint=0712.3256 |class=math.PR |mode=cs2}} |
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*{{Cite arXiv}} |
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*{{Citation|last=Lawler|first=Gregory F.|author-link=Gregory Lawler|title=Stochastic Loewner Evolution|url=http://www.math.cornell.edu/%7Elawler/encyclopedia.ps}} |
*{{Citation|last=Lawler|first=Gregory F.|author-link=Gregory Lawler|title=Stochastic Loewner Evolution|url=http://www.math.cornell.edu/%7Elawler/encyclopedia.ps|accessdate=2020-02-11|archive-date=2016-03-04|archive-url=https://web.archive.org/web/20160304044007/http://www.math.cornell.edu/~lawler/encyclopedia.ps|dead-url=no}} |
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*{{Citation|last=Lawler|first=Gregory F.|author-link=Gregory Lawler|title=Conformal invariance and 2D statistical physics|journal=Bull. Amer. Math. Soc.|volume=46|year=2009|pages=35–54|doi=10.1090/S0273-0979-08-01229-9}} |
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*{{Citation|last=Lawler|url=http://www.mrlonline.org/mrl/2001-008-004/2001-008-004-001.html|doi=10.4310/mrl.2001.v8.n4.a1|pages=401–411|number=4|volume=8|journal=Mathematical Research Letters|year=2001|mr=1849257|title=The dimension of the planar Brownian frontier is 4/3|first=Gregory F.|authorlink3=Wendelin Werner|first3=Wendelin|last3=Werner|authorlink2=Oded Schramm|first2=Oded|last2=Schramm|author-link=Gregory Lawler|arxiv=math/0010165}} |
*{{Citation|last=Lawler|url=http://www.mrlonline.org/mrl/2001-008-004/2001-008-004-001.html|doi=10.4310/mrl.2001.v8.n4.a1|pages=401–411|number=4|volume=8|journal=Mathematical Research Letters|year=2001|mr=1849257|title=The dimension of the planar Brownian frontier is 4/3|first=Gregory F.|authorlink3=Wendelin Werner|first3=Wendelin|last3=Werner|authorlink2=Oded Schramm|first2=Oded|last2=Schramm|author-link=Gregory Lawler|arxiv=math/0010165|accessdate=2020-02-11|archive-date=2019-09-08|archive-url=https://web.archive.org/web/20190908141240/http://www.mrlonline.org/mrl/2001-008-004/2001-008-004-001.html|dead-url=no}} |
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*{{Citation|first=C.|last=Loewner|author-link=Charles Loewner|title=Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I|journal=Math. Ann.|volume=89|number=1–2|year=1923|pages=103–121|jfm=49.0714.01|doi=10.1007/BF01448091|url=http://dml.cz/bitstream/handle/10338.dmlcz/125927/MathBohem_118-1993-3_7.pdf}} |
*{{Citation|first=C.|last=Loewner|author-link=Charles Loewner|title=Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I|journal=Math. Ann.|volume=89|number=1–2|year=1923|pages=103–121|jfm=49.0714.01|doi=10.1007/BF01448091|url=http://dml.cz/bitstream/handle/10338.dmlcz/125927/MathBohem_118-1993-3_7.pdf|accessdate=2020-02-11|archive-date=2019-09-26|archive-url=https://web.archive.org/web/20190926155107/https://dml.cz/bitstream/handle/10338.dmlcz/125927/MathBohem_118-1993-3_7.pdf|dead-url=no}} |
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*{{Citation|last=Mandelbrot|first=Benoît|author-link=Benoît Mandelbrot|title=The Fractal Geometry of Nature|publisher=W. H. Freeman|isbn=978-0-7167-1186-5|year=1982|url=https://archive.org/details/fractalgeometryo00beno}} |
*{{Citation|last=Mandelbrot|first=Benoît|author-link=Benoît Mandelbrot|title=The Fractal Geometry of Nature|publisher=W. H. Freeman|isbn=978-0-7167-1186-5|year=1982|url=https://archive.org/details/fractalgeometryo00beno}} |
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*{{Citation|title=Introduction to Schramm–Loewner evolutions|url=http://www.statslab.cam.ac.uk/~james/Lectures/sle.pdf|first=J. R.|last=Norris|author-link=James R. Norris|year=2010}} |
*{{Citation|title=Introduction to Schramm–Loewner evolutions|url=http://www.statslab.cam.ac.uk/~james/Lectures/sle.pdf|first=J. R.|last=Norris|author-link=James R. Norris|year=2010|accessdate=2020-02-11|archive-date=2019-07-14|archive-url=https://web.archive.org/web/20190714025509/http://www.statslab.cam.ac.uk/~james/Lectures/sle.pdf|dead-url=no}} |
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*{{Citation|last=Pommerenke|first=Christian|author-link=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series=Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher=Vandenhoeck & Ruprecht|year=1975}} (Chapter 6 treats the classical theory of Loewner's equation) |
*{{Citation|last=Pommerenke|first=Christian|author-link=Christian Pommerenke|title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen|series=Studia Mathematica/Mathematische Lehrbücher|volume=15|publisher=Vandenhoeck & Ruprecht|year=1975}} (Chapter 6 treats the classical theory of Loewner's equation) |
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*{{Citation|last=Schramm|first=Oded|author-link=Oded Schramm|title=Scaling limits of loop-erased random walks and uniform spanning trees|arxiv=math.PR/9904022|mr=1776084|year=2000|journal=Israel Journal of Mathematics|volume=118|pages=221–288|doi=10.1007/BF02803524}} Schramm's original paper, introducing SLE |
*{{Citation|last=Schramm|first=Oded|author-link=Oded Schramm|title=Scaling limits of loop-erased random walks and uniform spanning trees|arxiv=math.PR/9904022|mr=1776084|year=2000|journal=Israel Journal of Mathematics|volume=118|pages=221–288|doi=10.1007/BF02803524}} Schramm's original paper, introducing SLE |
2024年9月2日 (一) 00:49的最新版本
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
[编辑]- Uniform spanning tree, 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
[编辑]- 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
[编辑]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
[编辑]- 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
[编辑]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
[编辑]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
[编辑]- ^ 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
[编辑]- 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
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- Lawler, Gregory F., Stochastic Loewner Evolution, [2020-02-11], (原始内容存档于2016-03-04)
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- 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
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- 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
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