The growth exponent for planar loop-erased random walk

Robert Masson (University of British Columbia)


We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two dimensional discrete lattice.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1012-1073

Publication Date: May 17, 2009

DOI: 10.1214/EJP.v14-651


  1. Himanshu Agrawal and Deepak Dhar. Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions. Physical Review E, 63:no. 056115, 2001. Math. Review number not available.
  2. Martin Barlow and Robert Masson. Second moment estimates for loop-erased random walk. In preparation.
  3. Vincent Beffara. The dimension of the SLE curves. Ann. Probab., 36(4):1421-1452, 2008. MR2435854
  4. Federico Camia and Charles M. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys., 268(1):1-38, 2006. MR2249794
  5. Federico Camia and Charles M. Newman. Critical percolation exploration path and SLE6: a proof of convergence. Probab. Theory Related Fields, 139(3-4):473-519, 2007. MR2322705
  6. Deepak Dhar. The abelian sandpile and related models. Physica A, 263(4):4-25, 1999. Math. Review number not available.
  7. Richard Kenyon. The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2):239-286, 2000. MR1819995
  8. Harry Kesten. Hitting probabilities of random walks on Zd. Stochastic Process. Appl., 25(2):165-184, 1987. MR0915132
  9. Gregory F. Lawler. A self-avoiding random walk. Duke Math. J., 47(3):655-693, 1980. MR0587173
  10. Gregory F. Lawler. Intersections of random walks. Probability and its Applications. Birkhäuser Boston Inc., Boston, MA, 1991. MR1117680
  11. Gregory F. Lawler. The logarithmic correction for loop-erased walk in four dimensions. In Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993), number Special Issue, pages 347-361, 1995. MR1364896
  12. Gregory F. Lawler. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab., 1:no. 2, approx. 20 pp. (electronic), 1996. MR1386294
  13. Gregory F. Lawler. Loop-erased random walk. In Perplexing problems in probability, volume 44 of Progr. Probab., pages 197-217. Birkhäuser Boston, Boston, MA, 1999. MR1703133
  14. Gregory F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. MR2129588
  15. Gregory F. Lawler and Vlada Limic. The Beurling estimate for a class of random walks. Electron. J. Probab., 9:no. 27, 846-861 (electronic), 2004. MR2110020
  16. Gregory F. Lawler and Vlada Limic. Random walk: a modern introduction. To be published by Cambridge University Press.
  17. Gregory F. Lawler and Emily E. Puckette. The intersection exponent for simple random walk. Combin. Probab. Comput., 9(5):441-464, 2000. MR1810151
  18. Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187(2):275-308, 2001. MR1879851
  19. Gregory F. Lawler, Oded Schramm, and Wendelin Werner. One-arm exponent for critical 2D percolation. Electron. J. Probab., 7:no. 2, 13 pp. (electronic), 2002. MR1887622
  20. Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32(1B):939-995, 2004. MR2044671
  21. Gregory F. Lawler, Oded Schramm, and Wendelin Werner. On the scaling limit of planar self-avoiding walk. In Fractal geometry and applications: a jubilee of Benoî t Mandelbrot, Part 2, volume 72 of Proc. Sympos. Pure Math., pages 339-364. Amer. Math. Soc., Providence, RI, 2004. MR2112127
  22. Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883-924, 2005. MR2153402
  23. Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221-288, 2000. MR1776084
  24. Oded Schramm and Scott Sheffield. Harmonic explorer and its convergence to SLE4. Ann. Probab., 33(6):2127-2148, 2005. MR2184093
  25. Oded Schramm and Scott Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. arXiv:math/0605337, 2006. Math. Review number not available.
  26. Stanislav Smirnov. Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. arXiv:0708.0039, to appear in Ann. Math. Math. Review number not available.
  27. Stanislav Smirnov. Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math., 333(3):239-244, 2001. MR1851632
  28. Stanislav Smirnov and Wendelin Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett., 8(5-6):729-744, 2001. MR1879816
  29. Wendelin Werner. Random planar curves and Schramm-Loewner evolutions. In Lectures on probability theory and statistics, volume 1840 of Lecture Notes in Math., pages 107-195. Springer, Berlin, 2004. MR2079672
  30. David Bruce Wilson. Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pages 296-303, New York, 1996. ACM. MR1427525

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.