Lecture Notes in Mathematics 1840 Editors: J.--M.Morel, Cachan F. Takens, Groningen B. Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Boris Tsirelson Wendelin Werner Lectures on Probability Theory and Statistics Ecole d’EtedeProbabilit´ es´ de Saint-Flour XXXII - 2002 Editor: Jean Picard 13 Authors Editor Boris Tsirelson Jean Picard School of Mathematics Laboratoire de Mathematiques´ Appliquees´ TelAvivUniversity UMR CNRS 6620 Tel Av iv 69978 Universite´ Blaise Pascal Clermont-Ferrand Israel 63177 Aubiere` Cedex, France e-mail: [email protected] e-mail: [email protected] Wendelin Werner Laboratoire de Mathematiques´ Universite´ Paris-Sud Batˆ 425, 91405 Orsay Cedex France e-mail: [email protected] Cover picture: Blaise Pascal (1623-1662) Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2001): 60-01, 60Gxx, 60J65, 60K35, 82B20, 82b27, 82B41 ISSN 0075-8434 Lecture Notes in Mathematics ISSN 0721-5363 Ecole d’Ete´ des Probabilites´ de St. Flour ISBN 3-540-21316-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. 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Typesetting: Camera-ready TEXoutputbytheauthors SPIN: 10994733 41/3142/du - 543210 - Printed on acid-free paper Preface Three series of lectures were given at the 32nd Probability Summer School in Saint-Flour (July 7–24, 2002), by Professors Pitman, Tsirelson and Werner. In order to keep the size of the volume not too large, we have decided to split the publication of these courses into two parts. This volume contains the courses of Professors Tsirelson and Werner. The course of Professor Pitman, entitled “Combinatorial stochastic processes”, is not yet ready. We thank the authors warmly for their important contribution. 76 participants have attended this school. 33 of them have given a short lecture. The lists of participants and of short lectures are enclosed at the end of the volume. Finally, we give the numbers of volumes of Springer Lecture Notes where previous schools were published. Lecture Notes in Mathematics 1971: vol 307 1973: vol 390 1974: vol 480 1975: vol 539 1976: vol 598 1977: vol 678 1978: vol 774 1979: vol 876 1980: vol 929 1981: vol 976 1982: vol 1097 1983: vol 1117 1984: vol 1180 1985/86/87: vol 1362 1988: vol 1427 1989: vol 1464 1990: vol 1527 1991: vol 1541 1992: vol 1581 1993: vol 1608 1994: vol 1648 1995: vol 1690 1996: vol 1665 1997: vol 1717 1998: vol 1738 1999: vol 1781 2000: vol 1816 2001: vol 1837 Lecture Notes in Statistics 1986: vol 50 2001: vol 179 Contents Part I Boris Tsirelson: Scaling Limit, Noise, Stability Introduction..................................................... 5 1 AFirstLook ................................................ 6 2 AbstractNonsenseoftheScalingLimit ......................... 17 3 ScalingLimitandIndependence................................ 28 4 Example:TheNoiseMadebyaPoissonSnake................... 44 5 Stability .................................................... 56 6 GeneralizingWienerChaos.................................... 67 7 Example:TheBrownianWebasaBlackNoise................... 83 8 Miscellany................................................... 93 References......................................................103 Index...........................................................105 Part II Wendelin Werner: Random Planar Curves and Schramm- Loewner Evolutions 1 Introduction.................................................113 2 LoewnerChains..............................................124 3 ChordalSLE.................................................130 4 ChordalSLEandRestriction ..................................137 5 SLEandtheBrownianFrontier................................143 6 RadialSLE..................................................151 7 SomeCriticalExponentsforSLE...............................156 8 BrownianExponents..........................................161 9 SLE,USTandLERW........................................169 10 SLEandCriticalPercolation...................................176 11 WhatIsMissing .............................................185 References......................................................190 List of Participants ............................................197 List of Short Lectures .........................................199 Part I Boris Tsirelson: Scaling Limit, Noise, Stability B. Tsirelson and W. Werner: LNM 1840, J. Picard (Ed.), pp. 1–106, 2004. c Springer-Verlag Berlin Heidelberg 2004 Scaling Limit, Noise, Stability Boris Tsirelson School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel [email protected] //www.tau.ac.il/~tsirel/ Summary. Linear functions of many independent random variables lead to classical noises (white, Poisson, and their combinations) in the scaling limit. Some singular stochastic flows and some models of oriented percolation involve very nonlinear functions and lead to nonclassical noises. Two examples are examined, Warren’s ‘noise made by a Poisson snake’ and the author’s ‘Brownian web as a black noise’. Classical noises are stable, nonclassical are not. A new framework for the scaling limit is proposed. Old and new results are presented about noises, stability, and spectral measures. 1AFirstLook.............................................. 6 1.1 TwoToyModels............................................ 6 1.2 OurLimitingProcedures..................................... 8 1.3 ExamplesofHighSymmetry.................................. 11 1.4 ExampleofLowSymmetry................................... 12 1.5 Trees,NotCubes............................................ 14 1.6 Sub-σ-fields................................................. 15 2 Abstract Nonsense of the Scaling Limit ................... 17 2.1 MoreonOurLimitingProcedures............................. 17 2.2 Coarse Probability Space: Definition and Simple Example . 20 2.3 GoodUseofJointCompactification ........................... 22 3 Scaling Limit and Independence .......................... 28 3.1 Product of Coarse Probability Spaces . 28 3.2 DyadicCase................................................ 30 3.3 ScalingLimitofFourier-WalshCoefficients ..................... 34 3.4 TheLimitingObject......................................... 37 3.5 TimeShift;Noise............................................ 41 4 Boris Tsirelson 4 Example: The Noise Made by a Poisson Snake ............ 44 4.1 ThreeDiscreteSemigroups:AlgebraicDefinition................. 44 4.2 TheThreeDiscreteSemigroups:Representation................. 45 4.3 Random Walks and Stochastic Flows in Discrete Semigroups . 45 4.4 ThreeContinuousSemigroups................................. 48 4.5 Convolution Semigroups in These Continuous Semigroups . 49 4.6 GettingDyadic ............................................. 50 4.7 ScalingLimit............................................... 51 4.8 Noises ..................................................... 52 4.9 ThePoissonSnake .......................................... 53 5 Stability .................................................. 56 5.1 DiscreteCase............................................... 56 5.2 ContinuousCase............................................ 59 5.3 Back to Discrete: Two Kinds of Stability . 65 6 Generalizing Wiener Chaos ............................... 67 6.1 First Chaos, Decomposable Processes, Stability . 67 6.2 HigherLevelsofChaos....................................... 70 6.3 AnOldQuestionofJacobFeldman............................ 75 6.4 BlackNoise................................................. 78 7 Example: The Brownian Web as a Black Noise ............ 83 7.1 ConvolutionSemigroupoftheBrownianWeb................... 83 7.2 SomeGeneralArguments..................................... 84 7.3 TheKeyArgument.......................................... 85 7.4 Remarks................................................... 89 7.5 ACombinatorialBy-product.................................. 90 8 Miscellany ................................................ 93 8.1 BeyondtheOne-DimensionalTime............................ 93 8.2 The‘WaveNoise’Approach .................................. 94 8.3 Groups,Semigroups,Kernels.................................. 96 8.4 AbstractNonsenseofLeJan-Raimond’sTheory................. 98 References .....................................................103 Index ..........................................................105 Scaling Limit, Noise, Stability 5 Introduction