The growth exponent for planar loop-erased random walk
Abstract
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
References
- 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.
- Martin Barlow and Robert Masson. Second moment estimates for loop-erased random walk. In preparation.
- Vincent Beffara. The dimension of the SLE curves. Ann. Probab., 36(4):1421-1452, 2008. MR2435854
- Federico Camia and Charles M. Newman. Two-dimensional critical percolation: the full scaling limit. Comm. Math. Phys., 268(1):1-38, 2006. MR2249794
- 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
- Deepak Dhar. The abelian sandpile and related models. Physica A, 263(4):4-25, 1999. Math. Review number not available.
- Richard Kenyon. The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2):239-286, 2000. MR1819995
- Harry Kesten. Hitting probabilities of random walks on Zd. Stochastic Process. Appl., 25(2):165-184, 1987. MR0915132
- Gregory F. Lawler. A self-avoiding random walk. Duke Math. J., 47(3):655-693, 1980. MR0587173
- Gregory F. Lawler. Intersections of random walks. Probability and its Applications. Birkhäuser Boston Inc., Boston, MA, 1991. MR1117680
- 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
- Gregory F. Lawler. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab., 1:no. 2, approx. 20 pp. (electronic), 1996. MR1386294
- 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
- Gregory F. Lawler. Conformally invariant processes in the plane, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. MR2129588
- 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
- Gregory F. Lawler and Vlada Limic. Random walk: a modern introduction. To be published by Cambridge University Press.
- Gregory F. Lawler and Emily E. Puckette. The intersection exponent for simple random walk. Combin. Probab. Comput., 9(5):441-464, 2000. MR1810151
- Gregory F. Lawler, Oded Schramm, and Wendelin Werner. Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187(2):275-308, 2001. MR1879851
- 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
- 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
- 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
- Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883-924, 2005. MR2153402
- Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221-288, 2000. MR1776084
- Oded Schramm and Scott Sheffield. Harmonic explorer and its convergence to SLE4. Ann. Probab., 33(6):2127-2148, 2005. MR2184093
- Oded Schramm and Scott Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. arXiv:math/0605337, 2006. Math. Review number not available.
- 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.
- 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
- Stanislav Smirnov and Wendelin Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett., 8(5-6):729-744, 2001. MR1879816
- 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
- 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

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