Advertisement

Communications in Mathematical Physics

, Volume 318, Issue 2, pp 307–354 | Cite as

On the Rate of Convergence of Loop-Erased Random Walk to SLE2

  • Christian Beneš
  • Fredrik Johansson Viklund
  • Michael J. Kozdron
Article

Abstract

We derive a rate of convergence of the Loewner driving function for a planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE2. The proof uses a new estimate of the difference between the discrete and continuous Green’s functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE2, we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.

Keywords

Random Walk Simple Random Walk Grid Domain Loewner Chain Loewner Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Auer, P.: Some hitting probabilities of random walks on Z 2. In: Berkes, I., Csáki, E., Révész, P. eds., Limit Theorems in Probability and Statistics, Volume 57 of Colloquia Mathematica Societatis János Bolyai. (Budapest, Hungary), 1990. Amsterdam: North-Holland, pp. 9–25Google Scholar
  2. 2.
    Beneš, C.: Some Estimates for Planar Random Walk and Brownian Motion. Preprint, 2006. http://arxiv.org/abs/math/0611127v1 [math.DR], 2006
  3. 3.
    Borovkov A.A.: On the rate of convergence for the invariance principle. Theory Prob. Appl. 18, 207–225 (1973)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chelkak D., Smirnov S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189, 515–580 (2012)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Csörgő M., Révész P.: Strong Approximations in Probability and Statistics. Academic Press, New York (1981)Google Scholar
  6. 6.
    Duminil-Copin, H., Smirnov, S.: Conformal invariance of lattice models. Preprint, 2011. http://arxiv.org/abs/1109.1549v4 [math.PR], 2012
  7. 7.
    Fukai Y., Uchiyama K.: Potential kernel for two-dimensional random walk. Ann. Prob. 24, 1979–1992 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Garnett, J.B., Marshall, D.E.: Harmonic Measure. New York: Cambridge University Press, 2005Google Scholar
  9. 9.
    Haeusler E.: An Exact Rate of Convergence in the Functional Central Limit Theorem for Special Martingale Difference Arrays. Z. Wahr. verw. Geb. 65, 523–534 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. New York: Academic Press, 1980Google Scholar
  11. 11.
    Johansson Viklund, F.: Convergence Rates for Loop-Erased Random Walk and other Loewner Curves. http://arxiv.org/abs/1205.5734v1 [math.PR], 2012
  12. 12.
    Kesten, H.: Relations Between Solutions to a Discrete and Continuous Dirichlet Problem. In: Durrett, R., Kesten, H. eds., Random Walks, Brownian Motion and Interacting Particle Systems, Volume 28 of Progress in Probability. Boston, MA: Birkhäuser, 1991, pp. 309–321Google Scholar
  13. 13.
    Komlós J., Major P., Tusnády G.: An Approximation of Partial Sums of Independent RV’s, and the Sample DF. II. Z. Wahr. verw. Geb. 34, 33–58, (1976)Google Scholar
  14. 14.
    Kozdron M.J., Lawler G.F.: Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk. Electron. J. Prob. 10, 1442–1467 (2005)MathSciNetGoogle Scholar
  15. 15.
    Lawler, G.F.: Intersections of Random Walks. Boston, MA: Birkhäuser, 1991Google Scholar
  16. 16.
    Lawler, G.F.: Conformally Invariant Processes in the Plane. Volume 114 of Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 2005Google Scholar
  17. 17.
    Lawler G.F., Schramm O., Werner W.: The Dimension of the Planar Brownian Frontier is 4/3. Math. Res. Lett. 8, 401–411 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents, I: Half-plane exponents. Acta Math. 187, 237–273 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents, II: Plane exponents. Acta Math. 187, 275–308 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents III: Two-sided exponents. Ann. Inst. H. Poincaré Prob. Stat. 38, 109–123 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Lawler G.F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Prob. 32, 939–995 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Pommerenke, C.: Boundary Behaviour of Conformal Maps, Volume 299 of Grundlehren der mathematischen Wissenschaften. New York: Springer-Verlag, 1992Google Scholar
  23. 23.
    Rohde S., Schramm O.: Basic properties of SLE. Ann. of Math. (2) 161, 883–924 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Schramm, O.: Conformally invariant scaling limits: an overview and a collection of problems. In: Sanz-Solé, M., Soria, J., Varona, J.L., Verdera, J. eds, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006. Volume I. Zurich, Switzerland: European Mathematical Society, 2007, pp. 513–543Google Scholar
  26. 26.
    Schramm O., Sheffield S.: Harmonic explorer and its convergence to SLE4. Ann. Prob. 33, 2127–2148 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Schramm O., Sheffield S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Schramm O., Wilson D.B.: SLE coordinate changes. New York J. Math. 11, 659–669 (2005)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Smirnov, S.: Critical percolation in the plane. http://arxiv.org/abs/0909.4499v1 [math.PR], 2009
  30. 30.
    Smirnov S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, 239–244 (2001)ADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Smirnov, S.: Discrete Complex Analysis and Probability. In: Bhatia, R. ed., Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010. Volume I. New Delhi: Hindustan Book Agency, 2010, pp. 595–621Google Scholar
  32. 32.
    Smirnov S.: Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Ann. of Math. (2) 172, 1435–1467 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christian Beneš
    • 1
  • Fredrik Johansson Viklund
    • 2
  • Michael J. Kozdron
    • 3
  1. 1.Department of MathematicsBrooklyn College of the City University of New YorkBrooklynUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of Mathematics & StatisticsUniversity of ReginaReginaCanada

Personalised recommendations