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2009

Scaling of loop-erased walks in 2 to 4 dimensions

Grassberger, Peter

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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 $n\sim N^{D/2}$.
In $d=3$ we again find a power law, but with an exponent different from the one found in the most precise previous simulations: $D = 1.6236\pm 0.0004$.

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P. Grassberger, Phys. Rev. E to be published (2009)
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