Article

Journal of Statistical Physics

, Volume 136, Issue 2, pp 399-404

First online:

Scaling of Loop-Erased Walks in 2 to 4 Dimensions

  • Peter GrassbergerAffiliated withJohn-von-Neumann Institute for ComputingDepartment of Physics and Astrophysics, University of Calgary Email author 

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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=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 nN D/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.

Keywords

Loop-erased walks Critical exponents Logarithmic corrections