Acta Mathematica

The scaling limit of loop-erased random walk in three dimensions

Gady Kozma

Full-text: Open access


We show that the scaling limit exists and is invariant under dilations and rotations. We give some tools that might be useful to show universality.

Article information

Acta Math., Volume 199, Number 1 (2007), 29-152.

Received: 2 September 2005
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2007 © Institut Mittag-Leffler


Kozma, Gady. The scaling limit of loop-erased random walk in three dimensions. Acta Math. 199 (2007), no. 1, 29--152. doi:10.1007/s11511-007-0018-8.

Export citation


  • A izenman, M. & B urchard, A., Hölder regularity and dimension bounds for random curves. Duke Math. J., 99 (1999), 419–453.
  • B arlow, M. T. & B ass, R. F., Brownian motion and harmonianalysis on Sierpinski carpets. Canad. J. Math., 51 (1999), 673–744.
  • B ass, R. F., Probabilistic Techniques in Analysis. Probability and its Applications. Springer, New York, 1995.
  • B enjamini, I., P emantle, R. & P eres, Y., Martin capacity for Markov chains. Ann. Probab., 23:3 (1995), 1332–1346.
  • B erger, N. & B iskup, M., Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields, 137 (2007), 83–120.
  • B ollobás, B. & R iordan, O., Percolation. Cambridge University Press, New York, 2006.
  • B rydges, D. C. & I mbrie, J. Z., Branched polymers and dimensional reduction. Ann. of Math., 158 (2003), 1019–1039.
  • B rydges, D. C. & S pencer, T., Self-avoiding walk in 5 or more dimensions. Comm. Math. Phys., 97 (1985), 125–148.
  • B urdzy, K. & L awler, G. F., Nonintersection exponents for Brownian paths. I. Existence and an invariance principle. Probab. Theory Related Fields, 84 (1990), 393–410.
  • — Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab., 18:3 (1990), 981–1009.
  • C arlen, E. A., K usuoka, S. & S troock, D. W., Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist., 23 (1987), 245–287.
  • C ranston, M. C. & M ountford, T. S., An extension of a result of Burdzy and Lawler. Probab. Theory Related Fields, 89 (1991), 487–502.
  • D e M asi, A., F errari, P. A., G oldstein, S. & W ick, W. D., An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys., 55 (1989), 787–855.
  • D elmotte, T., Inégalité de Harnack elliptique sur les graphes. Colloq. Math., 72 (1997), 19–37.
  • — Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana, 15 (1999), 181–232.
  • D erbez, E. & S lade, G., The scaling limit of lattice trees in high dimensions. Comm. Math. Phys., 193 (1998), 69–104.
  • D uplantier, B., Loop-erased self-avoiding walks in two dimensions: exact critical exponents and winding numbers. Phys. A, 191 (1992), 516–522.
  • D uplantier, B. & K won, K. H., Conformal invariance and intersections of random walks. Phys. Rev. Lett., 61:22 (1988), 2514–2517.
  • D voretzky, A., E rdős, P. & K akutani, S., Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged, 12 (1950), 75–81.
  • G rigor'yan, A., Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana, 10 (1994), 395–452.
  • G rigor'yan, A. & S aloff-C oste, L., Stability results for Harnack inequalities. Ann. Inst. Fourier (Grenoble), 55 (2005), 825–890.
  • G romov, M., Hyperbolic manifolds, groups and actions, in Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, NY, 1978), Ann. of Math. Stud., 97, pp. 183–213. Princeton Univ. Press, Princeton, NJ, 1981.
  • G uttmann, A. J. & B ursill, R. J., Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods. J. Stat. Phys., 59 (1990), 1–9.
  • H ara, T., van der H ofstad, R. & S lade, G., Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab., 31:1 (2003), 349–408.
  • H ara, T. & S lade, G., Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys., 147 (1992), 101–136.
  • H ebisch, W. & S aloff-C oste, L., On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble), 51 (2001), 1437–1481.
  • van der H ofstad, R. & S lade, G., Convergence of critical oriented percolation to super-Brownian motion above 4+1 dimensions. Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 413–485.
  • H olopainen, I. & S oardi, P. M., A strong Liouville theorem for p-harmonic functions on graphs. Ann. Acad. Sci. Fenn. Math., 22 (1997), 205–226.
  • J erison, D., The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J., 53 (1986), 503–523.
  • K anai, M., Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan, 37 (1985), 391–413.
  • K enyon, R., The asymptotic determinant of the discrete Laplacian. Acta Math., 185 (2000), 239–286.
  • — Long-range properties of spanning trees. J. Math. Phys., 41 (2000), 1338–1363.
  • K esten, H., Hitting probabilities of random walks on Zd. Stochastic Process. Appl., 25 (1987), 165–184.
  • K ozlov, S. M., The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk, 40 (1985), 61–120, 238 (Russian); English translation in Russian Math. Surveys, 40 (1985), 73–145.
  • K ozma, G., Scaling limit of loop erased random walk – a naïve approach. Preprint, 2002. arXiv:math.PR/0212338.
  • K ozma, G. & S chreiber, E., An asymptotic expansion for the discrete harmonic potential. Electron. J. Probab., 9:1 (2004), 1–17.
  • L awler, G. F., A self-avoiding random walk. Duke Math. J., 47 (1980), 655–693.
  • — Loop-erased self-avoiding random walk and the Laplacian random walk. J. Phys. A, 20:13 (1987), 4565–4568.
  • — Intersections of random walks with random sets. Israel J. Math., 65 (1989), 113–132.
  • Intersections of Random Walks. Probability and its Applications. Birkhäuser, Boston, MA, 1991.
  • — The logarithmic correction for loop-erased walk in four dimensions, in Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993). J. Fourier Anal. Appl., Special Issue (1995), pp. 347–361.
  • — Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab., 1:2 (1996), 20 pp.
  • — Cut times for simple random walk. Electron. J. Probab., 1:13 (1996), 24 pp.
  • — Strict concavity of the intersection exponent for Brownian motion in two and three dimensions. Math. Phys. Electron. J., 4:5 (1998), 67 pp.
  • — Loop-erased random walk, in Perplexing Problems in Probability, Progress in Probability, 44, pp. 197–217. Birkhäuser, Boston, MA, 1999.
  • — The Laplacian-b random walk and the Schramm–Loewner evolution. Preprint, 2005.
  • L awler, G. F. & P uckette, E. E., The intersection exponent for simple random walk. Combin. Probab. Comput., 9:5 (2000), 441–464.
  • L awler, G. F., S chramm, O. & W erner, W., Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math., 187 (2001), 237–273.
  • — Values of Brownian intersection exponents. II. Plane exponents. Acta Math., 187 (2001), 275–308.
  • — Analyticity of intersection exponents for planar Brownian motion. Acta Math., 189 (2002), 179–201.
  • — Sharp estimates for Brownian non-intersection probabilities, in In and Out of Equilibrium (Mambucaba, 2000), Progress in Probability, 51, pp. 113–131. Birkhäuser, Boston, MA, 2002.
  • — Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab., 32:1B (2004), 939–995.
  • — On the scaling limit of planar self-avoiding walk, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2, Proc. Sympos. Pure Math., 72, pp. 339–364. Amer. Math. Soc., Providence, RI, 2004.
  • M ajumdar, S. N., Exact fractal dimension of the loop-erased self-avoiding walk in two dimensions. Phys. Rev. Lett., 68:15 (1992), 2329–2331.
  • M athieu, P. & P iatnitski, A. L., Quenched invariance principle for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463:2085 (2007), 2287–2307.
  • N guyen, B. G. & Y ang, W. S., Gaussian limit for critical oriented percolation in high dimensions. J. Statist. Phys., 78 (1995), 841–876.
  • P emantle, R., Choosing a spanning tree for the integer lattice uniformly. Ann. Probab., 19:4 (1991), 1559–1574.
  • P ittet, C. & S aloff-C oste, L., A survey on the relationships between volume growth, isoperimetry, and the behavior of simple random walk on Cayley graphs, with examples. Preprint, 2001.
  • R ichardson, D., Random growth in a tessellation. Proc. Cambridge Philos. Soc., 74 (1973), 515–528.
  • R ogers, L. C. G. & W illiams, D., Diffusions, Markov Processes, and Martingales. Vol. 1. Probability and Mathematical Statistics. Wiley, Chichester, 1994.
  • S chramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118 (2000), 221–288.
  • S chramm, O. & S heffield, S., The harmonic explorer and its convergence to SLE(4). Preprint, 2003. arXiv:math.PR/0310210.
  • S idoravicius, V. & S znitman, A. S., Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Related Fields, 129 (2004), 219–244.
  • S lade, G., Lattice trees, percolation and super-Brownian motion, in Perplexing Problems in Probability, Progress in Probability, 44, pp. 35–51. Birkhäuser, Boston, MA, 1999.
  • S mirnov, S., Critical percolation in the plane. Preprint, 2001.
  • — Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Preprint, 2007. arXiv:0708.0039.
  • S mirnov, S. & W erner, W., Critical exponents for two-dimensional percolation. Math. Res. Lett., 8 (2001), 729–744.
  • S oardi, P. M., Potential Theory on Infinite Networks. Lecture Notes in Math., 1590. Springer, Berlin–Heidelberg, 1994.
  • V aropoulos, N. T., Isoperimetric inequalities and Markov chains. J. Funct. Anal., 63 (1985), 215–239.
  • W ilson, 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. 293–303. ACM, New York, 1996.