Journal of Statistical Physics

, Volume 136, Issue 2, pp 399–404

Scaling of Loop-Erased Walks in 2 to 4 Dimensions


DOI: 10.1007/s10955-009-9787-0

Cite this article as:
Grassberger, P. J Stat Phys (2009) 136: 399. doi:10.1007/s10955-009-9787-0


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 

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