Skip to main content
SpringerLink
Log in

The Length of an SLE—Monte Carlo Studies

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm–Loewner evolution (SLE) for a suitable value of the parameter κ. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that if this fractal variation is used to parametrize the SLE, then the parametrized curves have the same distribution as the curves in the scaling limit of the lattice models with their natural parametrization.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauer, M., Bernard, D., Houdayer, J.: Dipolar SLE’s. J. Stat. Mech. 0503, P001 (2005). Archived as math-ph/0411038 in arXiv.org

    Google Scholar 

  2. Beffara, V.: Hausdorff dimensions for SLE6. Ann. Probab. 32, 2606–2629 (2004). Archived as math.PR/0204208 in arXiv.org

    Article  MATH  Google Scholar 

  3. Beffara, V.: The dimension of the SLE curves. Preprint. Archived as math.PR/0211322 in arXiv.org

  4. Camia, F., Newman, C.M.: Critical percolation exploration path and SLE(6): a proof of convergence. Preprint. Archived as math.PR/0604487 in arXiv.org

  5. Dudley, R.M.: Sample functions of the Gaussian process. Ann. Probab. 1, 66–103 (1973)

    MATH  Google Scholar 

  6. Fernández de la Vega, W.: On almost sure convergence of quadratic Brownian variation. Ann. Probab. 2, 551–552 (1974)

    MATH  Google Scholar 

  7. Kennedy, T.: A faster implementation of the pivot algorithm for self-avoiding walks. J. Stat. Phys. 106, 407–429 (2002). Archived as cond-mat/0109308 in arXiv.org

    Article  MATH  Google Scholar 

  8. Kennedy, T.: Monte Carlo tests of stochastic Loewner evolution predictions for the 2D self-avoiding walk. Phys. Rev. Lett. 88, 130601 (2002). Archived as math.PR/0112246 in arXiv.org

    Article  ADS  Google Scholar 

  9. Kennedy, T.: Conformal invariance and stochastic Loewner evolution predictions for the 2D self-avoiding walk—Monte Carlo tests. J. Stat. Phys. 114, 51–78 (2004). Archived as math.PR/0207231 in arXiv.org

    Article  MATH  Google Scholar 

  10. Kennedy, T.: A fast algorithm for simulating the chordal Schramm–Loewner evolution. J. Stat. Phys. (2007, in press). Archived as math.PR/0508002 in arXiv.org

  11. Kennedy, T.: Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves. Preprint. Archived as math.PR/0510604 in arXiv.org

  12. Lawler, G.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. Am. Math. Soc., Providence (2005)

    MATH  Google Scholar 

  13. Lawler, G.: Dimension and natural parametrization for SLE curves. Preprint (2007)

  14. Lawler, G., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Fractal Geometry and Applications: a Jubilee of Benoit Mandelbrot, Part 2. Proc. Sympos. Pure Math., vol. 72, pp. 339–364. Am. Math. Soc., Providence (2004). Archived as math.PR/0204277 in arXiv.org

    Google Scholar 

  15. Lawler, G., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004). Archived as math.PR/0112234 in arXiv.org

    Article  MATH  Google Scholar 

  16. Lévy, P.: Le mouvement brownien plan. Am. J. Math. 62, 487–550 (1940)

    Article  MATH  Google Scholar 

  17. Madras, N., Slade, G.: The Self-Avoiding Walk. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  18. Rohde, S.: Private communication (2005)

  19. Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000). Archived as math.PR/9904022 in arXiv.org

    MATH  Google Scholar 

  20. Smirnov, S.: Critical percolation in the plane. C.R. Acad. Sci. Paris Sér. I Math. 333, 239 (2001)

    MATH  Google Scholar 

  21. Smirnov, S.: Towards conformal invariance of 2D lattice models. In: Proceedings of the International Congress of Mathematicians (Madrid, 2006). European Mathematical Society

  22. Smirnov, S.: Conformal invariance in the Ising model. In: KITP, Santa Barbara, September 14, 2006. Available at http://online.kitp.ucsb.edu/online/sle06/smirnov/

  23. Wolff, U.: Comparison between cluster Monte Carlo algorithms in the Ising model. Phys. Lett. B228, 379–382 (1989)

    ADS  Google Scholar 

  24. Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Preprint. Archived as math.PR/0610304 in arXiv.org

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Tom Kennedy

Authors
  1. Tom Kennedy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tom Kennedy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kennedy, T. The Length of an SLE—Monte Carlo Studies. J Stat Phys 128, 1263–1277 (2007). https://doi.org/10.1007/s10955-007-9375-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-007-9375-0

Keywords

Search

Navigation