The Annals of Probability

Convergence rates for loop-erased random walk and other Loewner curves

Fredrik Johansson Viklund

Full-text: Open access

Abstract

We estimate convergence rates for curves generated by Loewner’s differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski’s boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm–Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial $\mathrm{SLE}_{2}$ path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be Hölder continuous in the capacity parameterization, assuming its driving term is Hölder continuous. We also briefly discuss the case when the curves are a priori known to be Hölder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.

Article information

Source
Ann. Probab., Volume 43, Number 1 (2015), 119-165.

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801554

Digital Object Identifier
doi:10.1214/13-AOP872

Mathematical Reviews number (MathSciNet)
MR3298470

Zentralblatt MATH identifier
1306.60118

Subjects
Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 30C35: General theory of conformal mappings

Keywords
Schramm–Loewner evolution loop-erased random walk Loewner equation

Citation

Johansson Viklund, Fredrik. Convergence rates for loop-erased random walk and other Loewner curves. Ann. Probab. 43 (2015), no. 1, 119--165. doi:10.1214/13-AOP872. https://projecteuclid.org/euclid.aop/1415801554


Export citation

References

  • [1] Aizenman, M. and Burchard, A. (1999). Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 419–453.
  • [2] Becker, J. and Pommerenke, C. (1982). Hölder continuity of conformal mappings and nonquasiconformal Jordan curves. Comment. Math. Helv. 57 221–225.
  • [3] Beneš, C. (2008). Counting planar random walk holes. Ann. Probab. 36 91–126.
  • [4] Beneš, C., Johansson Viklund, F. and Kozdron, M. J. (2013). On the rate of convergence of loop-erased random walk to $\mathrm{SLE}_{2}$. Comm. Math. Phys. 318 307–354.
  • [5] Garnett, J. B. and Marshall, D. E. (2008). Harmonic Measure. Cambridge Univ. Press, Cambridge.
  • [6] Johansson Viklund, F. and Lawler, G. F. (2011). Optimal Hölder exponent for the SLE path. Duke Math. J. 159 351–383.
  • [7] Johansson Viklund, F. and Lawler, G. F. (2012). Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math. 209 265–322.
  • [8] Johansson Viklund, F., Rohde, S. and Wong, C. (2014). On the continuity of $\mathrm{SLE}_{\kappa}$ in $\kappa$. Probab. Theory Related Fields 159 413–433.
  • [9] Kemppainen, A. and Smirnov, S. (2009). Random curves, scaling limits and Loewner evolutions. Unpublished manuscript.
  • [10] Lawler, G. F. (2005). Conformally Invariant Processes in the Plane. Amer. Math. Soc., Providence, RI.
  • [11] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Univ. Press, Cambridge.
  • [12] Lawler, G. F. and Puckette, E. E. (1997). The disconnection exponent for simple random walk. Israel J. Math. 99 109–121.
  • [13] Lawler, G. F., Schramm, O. and Werner, W. (2004). Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 939–995.
  • [14] Lind, J., Marshall, D. E. and Rohde, S. (2010). Collisions and spirals of Loewner traces. Duke Math. J. 154 527–573.
  • [15] Lind, J. and Rohde, S. (2012). Spacefilling curves and phases of the Loewner equation. Indiana Univ. Math. J. 61 2231–2249.
  • [16] Marshall, D. E. and Rohde, S. (2005). The Loewner differential equation and slit mappings. J. Amer. Math. Soc. 18 763–778 (electronic).
  • [17] Näkki, R. and Palka, B. (1982/83). Lipschitz conditions, $b$-arcwise connectedness and conformal mappings. J. Anal. Math. 42 38–50.
  • [18] Näkki, R. and Palka, B. (1986). Extremal length and Hölder continuity of conformal mappings. Comment. Math. Helv. 61 389–414.
  • [19] Pommerenke, Ch. (1966). On the Loewner differential equation. Michigan Math. J. 13 435–443.
  • [20] Pommerenke, Ch. (1992). Boundary Behaviour of Conformal Maps. Springer, Berlin.
  • [21] Rohde, S. and Schramm, O. (2005). Basic properties of SLE. Ann. of Math. (2) 161 883–924.
  • [22] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [23] Schramm, O. (2007). Conformally Invariant Scaling Limits: An Overview and a Collection of Problems. Eur. Math. Soc., Zürich.
  • [24] Schramm, O. and Wilson, D. B. (2005). SLE coordinate changes. New York J. Math. 11 659–669 (electronic).
  • [25] Smith, W. and Stegenga, D. A. (1987). A geometric characterization of Hölder domains. J. Lond. Math. Soc. (2) 35 471–480.
  • [26] Warschawski, S. E. (1950). On the degree of variation in conformal mapping of variable regions. Trans. Amer. Math. Soc. 69 335–356.
  • [27] Wong, C. W. C. (2014). Smoothness of Loewner slits. Trans. Amer. Math. Soc. 366 1475–1496.