Probability Surveys

Scaling limits and the Schramm-Loewner evolution

Gregory F. Lawler

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Abstract

These notes are from my mini-courses given at the PIMS summer school in 2010 at the University of Washington and at the Cornell probability summer school in 2011. The goal was to give an introduction to the Schramm-Loewner evolution to graduate students with background in probability. This is not intended to be a comprehensive survey of SLE.

Article information

Source
Probab. Surveys, Volume 8 (2011), 442-495.

Dates
First available in Project Euclid: 30 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.ps/1325264818

Digital Object Identifier
doi:10.1214/11-PS189

Mathematical Reviews number (MathSciNet)
MR2861136

Zentralblatt MATH identifier
1245.60078

Citation

Lawler, Gregory F. Scaling limits and the Schramm-Loewner evolution. Probab. Surveys 8 (2011), 442--495. doi:10.1214/11-PS189. https://projecteuclid.org/euclid.ps/1325264818


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References

  • [1] A. Belavin, A. Polyakov, A. Zamolodchikov (1984). Infinite conformal symmetry of critical fluctuations in two dimensions, J. Stat. Phys. 34, 763–774.
  • [2] A. Belavin, A. Polyakov, A. Zamolodchikov (1984). Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys B 241, 333–380.
  • [3] V. Beffara (2008). The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452.
  • [4] J. Cardy (1992). Critical percolation in finite geometries, J. Phys. A 25, L201–L206.
  • [5] J. Conway (1978, 1995). Functions of One Complex Variable I and II, Springer-Verlag.
  • [6] P. Duren (2001). Univalent Functions. Springer-Verlag.
  • [7] G. Lawler (2005). Conformally Invariant Processes in the Plane, Amer. Math. Soc.
  • [8] G. Lawler (2009). Schramm-Loewner evolution, in Statistical Mechanics, S. Sheffield and T. Spencer, ed., IAS/Park City Mathematical Series, AMS, 231–295.
  • [9] G. Lawler (2009). Partition functions, loop measure, and versions of SLE, J. Stat. Phys. 134, 813–837.
  • [10] G. Lawler, Defining SLE in multiply connected domains with the loop measure, in preparation.
  • [11] G. Lawler, V. Limic (2010), Random Walk: A Modern Introduction, Cambridge Univ. Press.
  • [12] G. Lawler, O. Schramm, W. Werner (2004), Conformal invariance of planar loop-erased random walks and uniform spanning trees, Annals of Probab. 32, 939–995.
  • [13] G. Lawler, O. Schramm. and W. Werner (2003). Conformal restriction: the chordal case. J. Amer. Math. Soc. 16, 917–955.
  • [14] G. Lawler, J. A. Trujillo Ferreras (2007). Random walk loop soup. Trans. Amer. Math. Soc. 359, 767–787.
  • [15] G. Lawler, W. Werner (2004), The Brownian loop soup, Probab. Theory Related Fields 128, 565–588.
  • [16] G. Lawler, M. Rezaei, in preparation.
  • [17] G. Lawler, S. Sheffield (2011). A natural parametrization for the Schramm-Loewner evolution, Annals of Probab. 39, 1896–1937.
  • [18] G. Lawler, B. Werness, Multi-point Green’s functions for SLE and an estimate of Beffara, to appear in Annals of Probab.
  • [19] G. Lawler, W. Zhou, SLE curves and natural parametrization, preprint.
  • [20] D. Marshall, S. Rohde (2005), The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18, 763–778.
  • [21] S. Rohde, O. Schramm (2005), Basic properties of SLE. Ann. of Math. 161, 883–924.
  • [22] O. Schramm (2000). Scaling limits of loop-erased random walk and uniform spanning trees, Israel J. Math. 118, 221-288.
  • [23] O. Schramm (2001). A percolation formula. Electron Comm. Probab. 6, 115–120.
  • [24] D. Wilson (1996). Generating random spanning trees more quickly than the cover time, Proc. STOC96, 296–303.
  • [25] D. Zhan (2007), Reversibility of chordal SLE, Annals of Prob. 36, 1472–1494.