Skip to main content
SpringerLink
Log in

Regularity and Convergence of Random Curves

  • Chapter
  • First Online:
Schramm–Loewner Evolution

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 24))

  • 871 Accesses

Abstract

In this chapter, we study regularity properties curves in the capacity parametrization and their convergence with respect to the uniform norm on compact time intervals.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Notes

  1. 1.

    It is possible to give a more quantitative estimates for this convergence using the harmonic measure.

  2. 2.

    Remember that constant driving term corresponds to the trivial Loewner chain, which is a straight vertical line segment.

  3. 3.

    A boundary point is accessible, if there is a Jordan arc in the domain ending at that point. If we apply a conformal map from the domain onto \(\mathbb {D}\), say, then the image of that arc is continuous up to the boundary. Consequently, the accessible point is always a limit along the image of a Jordan arc in \(\mathbb {D}\) under a conformal map from \(\mathbb {D}\) onto the domain. The radial limit of the conformal map at the same boundary point of \(\mathbb {D}\) exists and is equal to the other limit as follows from Corollary 2.17 of [9].

  4. 4.

    A careful reader can notice that \(E^*\) defined in the latter way, which is more general, doesn’t necessarily define a (simple) graph, but a multigraph where a pair of vertices can be linked by several edges and where the endpoints of an edge don’t need to be distinct vertices.

  5. 5.

    It is natural to select the edge that belongs to \(\partial \varOmega _\delta \) if that exists. If it doesn’t exist, we can always add such an edge to \(\partial \varOmega _\delta \) without disturbing any of the required properties.

  6. 6.

    Such a curve is called a crossing and a crossing that doesn’t contain a proper subcrossing is called a minimal crossing .

  7. 7.

    A small calculation shows that the long side of the rhombi has length \((2m-1)n\) and the short side \((m-1)n\).

  8. 8.

    The minimum number of crossings is finite since there are even smooth crossings such as any “radial” path \(t \mapsto \phi (z_0 + t e^{i \theta })\), \(t \in (r,R)\) and \(\theta \in \mathbb {R}\).

  9. 9.

    That is, define \([\gamma ] = \{ \gamma \circ \phi \,:\, \phi : [0,1] \rightarrow [0,1] \text { increasing homeomorphism}\}\).

References

  1. Aizenman, M., Burchard, A.: Hölder regularity and dimension bounds for random curves. Duke Math. J. 99(3), 419–453 (1999). https://doi.org/10.1215/S0012-7094-99-09914-3

  2. Beffara, V.: Cardy’s formula on the triangular lattice, the easy way. In: Universality and Renormalization, Fields Inst. Commun, vol. 50, pp. 39–45. American Mathematical Society, Providence, RI (2007)

    Google Scholar 

  3. Grimmett, G.: Percolation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 321, 2nd edn. Springer-Verlag, Berlin (1999). https://doi.org/10.1007/978-3-662-03981-6

  4. Grimmett, G.: Probability on Graphs, Institute of Mathematical Statistics Textbooks, vol. 1. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/CBO9780511762550. Random processes on graphs and lattices

  5. Johansson Viklund, F., Rohde, S., Wong, C.: On the continuity of \(\text{SLE}_\kappa \) in \(\kappa \). Probab. Theory Relat. Fields 159(3–4), 413–433 (2014). https://doi.org/10.1007/s00440-013-0506-z

  6. Kemppainen, A., Smirnov, S.: Random curves, scaling limits and Loewner evolutions. Ann. Probab. 45(2), 698–779 (2017). https://doi.org/10.1214/15-AOP1074

  7. Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence, RI (2005)

    Google Scholar 

  8. Lawler, G.F.: Fractal and multifractal properties of Schramm-Loewner evolution. In: Probability and Statistical Physics in Two and More Dimensions, Clay Math. Proc., vol. 15, pp. 277–318. American Mathematical Society, Providence, RI (2012)

    Google Scholar 

  9. Pommerenke, C.: Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer-Verlag, Berlin (1992). https://doi.org/10.1007/978-3-662-02770-7

  10. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad.Sci. Paris Sér. I Mathématique 333(3), 239–244 (2001). https://doi.org/10.1016/S0764-4442(01)01991-7

  11. Smirnov, S.: Critical percolation in the plane. arXiv.org (2009). Long version, arXiv:0909.4499

  12. Viklund, F.J., Lawler, G.F.: Optimal Hölder exponent for the SLE path. Duke Math. J. 159(3), 351–383 (2011). https://doi.org/10.1007/s00440-013-0506-z

  13. Werner, W.: Lectures on two-dimensional critical percolation. In: Statistical Mechanics, IAS/Park City Math. Ser., vol. 16, pp. 297–360. American Mathematical Society, Providence, RI (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Antti Kemppainen

Authors
  1. Antti Kemppainen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antti Kemppainen .

Rights and permissions

Reprints and permissions

Copyright information

© 2017 The Author(s)

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kemppainen, A. (2017). Regularity and Convergence of Random Curves. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_6

Download citation

Publish with us

Policies and ethics

Search

Navigation