Wendelin Werner: Random Planar Curves and Schramm-Loewner Evolutions

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1840)

Abstract

  • 1 Introduction
    • 1.1 General Motivation

    • 1.2 Loop-Erased Random Walks

    • 1.3 Iterations of Conformal Maps and SLE

    • 1.4 The Critical Percolation Exploration Process

    • 1.5 Chordal versus Radial

    • 1.6 Conclusion

  • 2 Loewner Chains
    • 2.1 Measuring the Size of Subsets of the Half-Plane

    • 2.2 Loewner Chains

  • 3 Chordal SLE
    • 3.1 Definition

    • 3.2 A First Computation

    • 3.3 Chordal \(SLE_\kappa \) in Other Domains

    • 3.4 Transience

  • 4 Chordal SLE and Restriction
    • 4.1 Image of SLE under Conformal Maps

    • 4.2 Locality for SLE6

    • 4.3 Restriction for SLE8/3

  • 5 SLE and the Brownian Frontier
    • 5.1 A Reflected Brownian Motion

    • 5.2 Brownian Excursions and SLE8/3

  • 6 Radial SLE
    • 6.1 Definitions

    • 6.2 Relation between Radial and Chordal SLE

    • 6.3 Radial SLE6 and Reflected Brownian Motion

  • 7 Some Critical Exponents for SLE
    • 7.1 Disconnection Exponents

    • 7.2 Derivative Exponents

    • 7.3 First Consequences

  • 8 Brownian Exponents
    • 8.1 Introduction

    • 8.2 Brownian Crossings

    • 8.3 Disconnection Exponent

    • 8.4 Other Exponents

    • 8.5 Hausdorff Dimensions

  • 9 SLE, UST and LERW
    • 9.1 Introduction, LERW

    • 9.2 Uniform Spanning Trees, Wilson’s Algorithm

    • 9.3 Convergence to Chordal SLE8

    • 9.4 The Loop-Erased Random Walk

  • 10 SLE and Critical Percolation
    • 10.1 Introduction

    • 10.2 The Cardy-Smirnov Formula

    • 10.3 Convergence to SLE6 and Consequences

  • 11 What Is Missing
    • 11.1 A List of Ideas

    • 11.2 A List of Open Problems

  • References

Mathematics Subject Classification (2000):

60-01 60Gxx 60J65 60K35 82B20 82b27 82B41 

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Copyright information

© Springer-Verlag Berlin/Heidelberg 2004

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Université Paris-Sud Orsay Cedex France

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