Advertisement

Journal of Statistical Physics

, Volume 132, Issue 4, pp 721–754 | Cite as

LERW as an Example of Off-Critical SLEs

  • Michel Bauer
  • Denis Bernard
  • Kalle Kytölä
Article

Abstract

Two dimensional loop erased random walk (LERW) is a random curve, whose continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter κ=2. In this article we study “off-critical loop erased random walks”, loop erasures of random walks penalized by their number of steps. On one hand we are able to identify counterparts for some LERW observables in terms of symplectic fermions (c=−2), thus making further steps towards a field theoretic description of LERWs. On the other hand, we show that it is possible to understand the Loewner driving function of the continuum limit of off-critical LERWs, thus providing an example of application of SLE-like techniques to models near their critical point. Such a description is bound to be quite complicated because outside the critical point one has a finite correlation length and therefore no conformal invariance. However, the example here shows the question need not be intractable. We will present the results with emphasis on general features that can be expected to be true in other off-critical models.

Keywords

Stochastic Loewner evolutions Near critical interfaces Loop erased random walks 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bauer, M., Bernard, D.: Conformal field theories of stochastic Loewner evolutions. Commun. Math. Phys. 239(3), 493–521 (2003) zbMATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Bauer, M., Bernard, D.: SLE, CFT and zig-zag probabilities. In: Proceedings of the Conference ‘Conformal Invariance and Random Spatial Processes’, Edinburgh, July 2003 Google Scholar
  3. 3.
    Bauer, M., Bernard, D.: 2D growth processes: SLE and Loewner chains. Phys. Rep. 432(3–4), 115–222 (2006) CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bauer, M., Bernard, D., Houdayer, J.: Dipolar SLEs. J. Stat. Mech. 0503, P001 (2005) Google Scholar
  5. 5.
    Bauer, M., Bernard, D., Kytölä, K.: Multiple Schramm-Loewner evolutions and statistical mechanics martingales. J. Stat. Phys. 120(5–6), 1125–1163 (2005) zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bauer, M., Bernard, D., Kennedy, T.G.: (2008, in preparation) Google Scholar
  7. 7.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984) zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Camia, F., Fontes, L., Newman, C.: The scaling limit geometry of near-critical 2d percolation. J. Stat. Phys. 125(5–6), 1155–1171 (2006). arXiv:cond-mat/0510740 CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Caracciolo, S., Jacobsen, J.L., Saleur, H., Sokal, A.D., Sportiello, A.: Fermionic field theory for trees and forests. Phys. Rev. Lett. 93, 080601 (2004) CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Cardy, J.: Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge (1996) Google Scholar
  11. 11.
    Cardy, J.: SLE for theoretical physicists. Ann. Phys. 318(1), 81–118 (2005) zbMATHCrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Dudley, R.M.: Real Analysis and Probability. Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002) zbMATHGoogle Scholar
  13. 13.
    Kager, W., Nienhuis, B.: A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115(5–6), 1149–1229 (2004) CrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. GTM, vol. 113. Springer, Berlin (1991) zbMATHGoogle Scholar
  15. 15.
    Kausch, H.-G.: Symplectic Fermions. Nucl. Phys. B 583, 513–541 (2000) zbMATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Kennedy, T.G.: The Length of an SLE—Monte Carlo Studies. J. Stat. Phys. 128(6), 1263–1277 (2007). arXiv:math/0612609v2 zbMATHCrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Kytölä, K.: On conformal field theory of SLE(kappa, rho). J. Stat. Phys. 123(6), 1169–1181 (2006) zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kytölä, K.: Virasoro module structure of local martingales of SLE variants. Rev. Math. Phys. 19(5), 455–509 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lawler, G.F.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence (2005) zbMATHGoogle Scholar
  20. 20.
    Lawler, G.F.: Dimension and natural parametrization for SLE curves. arXiv:0712.3263 (2007)
  21. 21.
    Lawler, G.F., Sheffield, S.: Construction of the natural parametrization for SLE curves (2008, in preparation) Google Scholar
  22. 22.
    Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lawler, G., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Amer. Math. Soc. 16(4), 917–955 (2003). (electronic) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Majumdar, S.: Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions. Phys. Rev. Lett. 68, 2329–2331 (1992) CrossRefADSGoogle Scholar
  26. 26.
    Makarov, N., Smirnov, S.: Massive SLEs (2008, in preparation) Google Scholar
  27. 27.
    Nolin, P., Werner, W.: Asymmetry of near-critical percolation interfaces. arXiv:0710.1470 (2007)
  28. 28.
    Öksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 5th ed. Springer Universitext. Springer, Berlin (2003) zbMATHGoogle Scholar
  29. 29.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  30. 30.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Schramm, O., Wilson, D.: SLE coordinate changes. N.Y. J. Math. 11, 659–669 (2005) zbMATHMathSciNetGoogle Scholar
  32. 32.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C.R. Acad. Sci. Paris 333, 239–244 (2001) zbMATHGoogle Scholar
  33. 33.
    Werner, W.: Random planar curves and Schramm-Loewner evolutions. In: Lectures on probability theory and statistics. Lecture Notes in Math., vol. 1840, pp. 107–195. Springer, Berlin (0000) Google Scholar
  34. 34.
    Wilson, D.B.: 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, pp. 296–303. ACM, New York (1996) CrossRefGoogle Scholar
  35. 35.
    Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36(2), 467–529 (2008) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Service de Physique Théorique de SaclayCEA-SaclayGif-sur-YvetteFrance
  2. 2.Laboratoire de Physique ThéoriqueEcole Normale SupérieureParisFrance
  3. 3.CNRS; Laboratoire de Physique ThéoriqueEcole Normale SupérieureParisFrance
  4. 4.Laboratoire de Physique Théorique et Modèles StatistiquesUniversité Paris SudOrsayFrance

Personalised recommendations