Electronic Journal of Probability

One-point function estimates for loop-erased random walk in three dimensions

Xinyi Li and Daisuke Shiraishi

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Abstract

In this work, we consider the loop-erased random walk (LERW) in three dimensions and give asymptotic estimates for the one-point function of LERW and the non-intersection probability of LERW and simple random walk for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 111, 46 pp.

Dates
Received: 27 July 2018
Accepted: 8 September 2019
First available in Project Euclid: 9 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1570586691

Digital Object Identifier
doi:10.1214/19-EJP361

Mathematical Reviews number (MathSciNet)
MR4017129

Zentralblatt MATH identifier
07142905

Subjects
Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Keywords
loop-erased random walk

Rights
Creative Commons Attribution 4.0 International License.

Citation

Li, Xinyi; Shiraishi, Daisuke. One-point function estimates for loop-erased random walk in three dimensions. Electron. J. Probab. 24 (2019), paper no. 111, 46 pp. doi:10.1214/19-EJP361. https://projecteuclid.org/euclid.ejp/1570586691


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References

  • [1] C. Beneš, G. F. Lawler, and F. Viklund. Scaling limit of the loop-erased random walk Green’s function. Probab. Theory Relat. Fields, 166(1-2):271–319, 2016.
  • [2] A. J. Guttmann and R. J. Bursill. Critical exponents for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys., 59(1):1–9, 1990.
  • [3] R. Kenyon. The asymptotic determinant of the discrete Laplacian. Acta Math., 185(2):239–286, 2000.
  • [4] G. Kozma. The scaling limit of loop-erased random walk in three dimensions. Acta Math., 199(1):29–152, 2007.
  • [5] G. F. Lawler. A self-avoiding random walk. Duke Math. J., 47(3):655–693, 1980.
  • [6] G. F. Lawler. The logarithmic correction for loop-erased walk in four dimensions. J. Fourier Anal. Appl., Special Issue, 347–362, 1995.
  • [7] G. F. Lawler. Cut times for simple random walk. Electron. J. Probab., 1(13):1–24, 1996.
  • [8] G. F. Lawler. Loop-erased random walk. In Perplexing problems in probability, pp. 197–217. Birkhäuser, Basel, 1999.
  • [9] G. F. Lawler. Conformally Invariant Processes in the Plane. American Mathematical Society, Providence, 2005.
  • [10] G. F. Lawler. Intersections of Random Walks. Springer, 2013.
  • [11] G. F. Lawler. Topics in loop measures and the loop-erased walk. Probab. Surveys, 15:28–101, 2018.
  • [12] G. F. Lawler. The infinite two-sided loop-erased random walk. Preprint, available at arXiv:1802.06667.
  • [13] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge University Press, Cambridge, 2010.
  • [14] G. F. Lawler, O. Schramm and W. Werner. Conformal invariance of planar loop-erased random walks and uniform spanning trees. In Selected Works of Oded Schramm, pp. 931–987. Springer, Berlin, 2011.
  • [15] X. Li and D. Shiraishi. Convergence of three-dimensional loop-erased random walk in the natural parametrization. Preprint, available at arXiv:1811.11685.
  • [16] R. Masson. The growth exponent for planar loop-erased random walk. Electron. J. Probab., 14:1012–1073, 2009.
  • [17] P. Mörters, Y. Peres. Brownian Motion. Cambridge University Press, Cambridge, 2010.
  • [18] R. Pemantle. Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19(4):1559–1574, 1991.
  • [19] A. Sapozhnikov and D. Shiraishi. On Brownian motion, simple paths, and loops. Prob. Theory Relat. Fields, 172(3-4):615–662, 2018.
  • [20] O. Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math., 118(1):221–288, 2000.
  • [21] D. Shiraishi. Growth exponent for loop-erased random walk in three dimensions. Ann. Probab., 46(2):687–774, 2018.
  • [22] D. Shiraishi. Hausdorff dimension of the scaling limit of loop-erased random walk in three dimensions. Ann. I. H. Poincaré Probab. Statist., 55(2):791–834, 2019.
  • [23] K. J. Wiese and A. A. Fedorenko. Field theories for loop-erased random walks. Preprint, available at arXiv:1802.08830.
  • [24] D. B. Wilson. Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pp. 296–303. ACM, New York 1996.
  • [25] D. B. Wilson. The dimension of loop-erased random walk in 3D. Phys. Rev. E, 82(6):062102, 2010.