English

Scaling of loop-erased walks in 2 to 4 dimensions

Statistical Mechanics 2015-05-13 v1

Abstract

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=4d=4, simulations in lower dimensions were done for completeness and in order to test the algorithm. In d=2d=2, we verify with high precision the prediction D=5/4D=5/4, where the number of steps nn after erasure scales with the number NN of steps before erasure as nND/2n\sim N^{D/2}. In d=3d=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±0.0004D = 1.6236\pm 0.0004. Finally, we see clear deviations from the naive scaling nNn\sim N in d=4d=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.

Keywords

Cite

@article{arxiv.0905.3440,
  title  = {Scaling of loop-erased walks in 2 to 4 dimensions},
  author = {Peter Grassberger},
  journal= {arXiv preprint arXiv:0905.3440},
  year   = {2015}
}

Comments

3 pages, including 3 figures