{"title":"平面临界渗流的保形不变尺度极限","authors":"Nike Sun","doi":"10.1214//11-PS180","DOIUrl":null,"url":null,"abstract":"This is an introductory account of the emergence of conformal \ninvariance in the scaling limit of planar critical percolation. We give \nan exposition of Smirnov's theorem (2001) on the conformal invariance \nof crossing probabilities in site percolation on the triangular \nlattice. We also give an introductory account of Schramm-Loewner \nevolutions (SLE ĸ ), a one-parameter family of conformally \ninvariant random curves discovered by Schramm (2000). The article is \norganized around the aim of proving the result, due to Smirnov (2001) \nand to Camia and Newman (2007), that the percolation exploration path \nconverges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2009-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"28","resultStr":"{\"title\":\"Conformally invariant scaling limits in planar critical percolation\",\"authors\":\"Nike Sun\",\"doi\":\"10.1214//11-PS180\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This is an introductory account of the emergence of conformal \\ninvariance in the scaling limit of planar critical percolation. We give \\nan exposition of Smirnov's theorem (2001) on the conformal invariance \\nof crossing probabilities in site percolation on the triangular \\nlattice. We also give an introductory account of Schramm-Loewner \\nevolutions (SLE ĸ ), a one-parameter family of conformally \\ninvariant random curves discovered by Schramm (2000). The article is \\norganized around the aim of proving the result, due to Smirnov (2001) \\nand to Camia and Newman (2007), that the percolation exploration path \\nconverges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.\",\"PeriodicalId\":46216,\"journal\":{\"name\":\"Probability Surveys\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2009-11-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"28\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Probability Surveys\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214//11-PS180\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Surveys","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214//11-PS180","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal
invariance in the scaling limit of planar critical percolation. We give
an exposition of Smirnov's theorem (2001) on the conformal invariance
of crossing probabilities in site percolation on the triangular
lattice. We also give an introductory account of Schramm-Loewner
evolutions (SLE ĸ ), a one-parameter family of conformally
invariant random curves discovered by Schramm (2000). The article is
organized around the aim of proving the result, due to Smirnov (2001)
and to Camia and Newman (2007), that the percolation exploration path
converges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.