Advertisement

Probability Theory and Related Fields

, Volume 172, Issue 3–4, pp 615–662 | Cite as

On Brownian motion, simple paths, and loops

  • Artem SapozhnikovEmail author
  • Daisuke Shiraishi
Article
  • 345 Downloads

Abstract

We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in \((1,\frac{5}{3}]\). We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results would give a new strategy to the existence of the scaling limit of the loop erased random walk and its rotational invariance.

Mathematics Subject Classification

60K40 60D05 60G50 60K35 

Notes

Acknowledgements

Enormous thanks go to Alain-Sol Sznitman for his helpful discussions, encouragements, and fruitful comments. We also thank Yinshan Chang and Wendelin Werner for useful discussions and suggestions, and Yinshan Chang for a careful reading of the paper. This project was carried out while the second author was enjoying the hospitality of the Forschungsinstitut für Mathematik of the ETH Zürich and the Max Planck Institute for Mathematics in the Sciences. He wishes to thank these institutions. The research of the second author has been supported by the Japan Society for the Promotion of Science (JSPS). Finally, the second author thanks Hidemi Aihara for all her understanding and support.

References

  1. 1.
    Ball, K., Sterbenz, J.: Explicit bounds for the return probability of simple random walks. J. Theor. Probab. 18(2), 317–326 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beffara, V.: The dimension of the SLE curves. Ann. Probab. 36(4), 1421–1452 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)Google Scholar
  4. 4.
    Camia, F.: Brownian loops and conformal fields. arXiv:1501.04861
  5. 5.
    Dvoretzky, A., Erdős, P., Kakutani, S.: Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged. 12, 75–81 (1950)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Fristedt, B.: An extension of a theorem of S. J. Taylor concerning the multiple points of the symmetric stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9, 62–64 (1967)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guttmann, A.J., Bursill, R.J.: Critical exponents for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys. 59(1), 1–9 (1990)CrossRefGoogle Scholar
  8. 8.
    Kakutani, S.: On Brownian motions in n-space. Proc. Imp. Acad. Tokyo 20, 648–652 (1944)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kenyon, R.: The asymptotic determinant of the discrete Laplacian. Acta Math. 185(2), 239–286 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kesten, H.: Hitting probabilities of random walks on \({\mathbb{Z}}^{d}\). Stoch. Process. Appl. 25, 165–184 (1987)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kozma, G.: The scaling limit of loop-erased random walk in three dimensions. Acta Math. 199(1), 29–152 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lawler, G.F.: Intersections of Random Walks. Birkhauser, Boston (1991)zbMATHGoogle Scholar
  13. 13.
    Lawler, G.F.: A self avoiding walk. Duke Math. J. 47, 655–694 (1980)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lawler, G.F.: Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1, 1–20 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lawler, G.F.: Loop-erased random walk. In: Bramson, M., Durrett, R. (eds.) Perplexing Problems in Probability. Progress in Probability, vol. 44, pp. 197–217. Birkhäuser Boston, Boston, MA (1999)CrossRefGoogle Scholar
  16. 16.
    Lawler, G.F.: The probability that planar loop-erased random walk uses a given edge. Electron. J. Probab. 19, 1–13 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Lawler, G.F., Limic, V.: Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)Google Scholar
  18. 18.
    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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal restriction: the chordal case. J. Am. Math. Soc. 16(4), 917–955 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lawler, G.F., Trujillo Ferreras, J.: Random walk loop soup. Trans. Am. Math. Soc. 359(2), 767–787 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Le Jan, Y.: Markov Paths, Loops and Fields. Lecture Notes in Mathematics, vol. 2026. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  23. 23.
    Lévy, P.: Le mouvement brownien plan. Am. J. Math. 62, 487–550 (1940)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Masson, R.: The growth exponent for planar loop-erased random walk. Electron. J. Probab. 14, 1012–1073 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Pemantle, R.: Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19(4), 1559–1574 (1991)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118(1), 221–288 (2000)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Shiraishi, D.: Growth exponent for loop-erased random walk in three dimensions. Ann. Probab. (to appear)Google Scholar
  28. 28.
    Shiraishi, D.: Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions. Preprint, arXiv:1604.08091
  29. 29.
    Slade, G.: Self-avoiding walks. Math. Intell. 16(1), 29–35 (1994)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Symanzik, K.: Euclidean quantum field theory. Scuola internazionale di Fisica “Enrico Fermi” XLV, 152–223 (1969)Google Scholar
  31. 31.
    Sznitman, A.-S.: Topics in Occupation Times and Gaussian Free Fields. Zürich Lectures in Advanced Mathematics. European Mathematical Society, Zürich (2012)CrossRefGoogle Scholar
  32. 32.
    Taylor, S.J.: Multiple points for the sample paths of the symmetric stable process. Zeitschr. Wahrsch. verw. Gebiete 5, 247–264 (1966)MathSciNetCrossRefGoogle Scholar
  33. 33.
    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)Google Scholar
  34. 34.
    Wilson, D.B.: The dimension of loop-erased random walk in 3D. Phys. Rev. E 82(6), 062102 (2010)CrossRefGoogle Scholar
  35. 35.
    Zhan, D.: Loop-erasure of plane Brownian motion. Communications in Mathematical Physics 303(3), 709–720 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeipzigLeipzigGermany
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan

Personalised recommendations