Journal of Statistical Physics

, Volume 136, Issue 2, pp 399–404

Scaling of Loop-Erased Walks in 2 to 4 Dimensions



We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2, 3 and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d=4, simulations in lower dimensions were done for completeness and in order to test the algorithm. In d=2, we verify with high precision the prediction D=5/4, where the number of steps n after erasure scales with the number N of steps before erasure as nND/2. In d=3 we again find a power law, but with an exponent different from the one found in the most precise previous simulations: D=1.6236±0.0004. Finally, we see clear deviations from the naive scaling nN in d=4. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.


Loop-erased walks Critical exponents Logarithmic corrections 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Majumdar, S.N.: Phys. Rev. Lett. 68, 2329 (1992) CrossRefADSGoogle Scholar
  2. 2.
    Duplantier, B.: Physica A 191, 516 (1992) CrossRefADSGoogle Scholar
  3. 3.
    Schramm, O.: Isr. J. Math. 118, 221 (2000) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Lawler, G.F.: Duke Math. J. 47, 655 (1980) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lawler, G.F.: J. Fourier Anal. Appl. 347 (1995). Special issue Google Scholar
  6. 6.
    Fedorenko, A.A., Le Doussal, P., Wiese, K.J.: J. Stat. Phys. 133, 805 (2008) MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Agrawal, H., Dhar, D.: Phys. Rev. E 63, 056115 (2001) CrossRefADSGoogle Scholar
  8. 8.
    Guttmann, A.J., Bursill, R.J.: J. Stat. Phys. 59, 1 (1990) CrossRefADSGoogle Scholar
  9. 9.
    Hsu, H.-P., Grassberger, P.: J. Stat. Mech., P01007 (2005) Google Scholar
  10. 10.
    Grassberger, P.: Phys. Rev. E 67, 036101 (2003) CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Grassberger, P.: Phys. Rev. E 79, 052104 (2009) CrossRefADSGoogle Scholar
  12. 12.
    Ziff, R.M.: Comput. Phys. 12, 385 (1998) CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.John-von-Neumann Institute for ComputingJülichGermany
  2. 2.Department of Physics and AstrophysicsUniversity of CalgaryAlbertaCanada

Personalised recommendations