Scaling of loop-erased walks in 2 to 4 dimensions

TL;DR

This paper investigates the scaling behavior of loop-erased random walks across dimensions 2 to 4 through simulations, confirming theoretical predictions in 2D and 4D, and providing new precise estimates in 3D.

Contribution

It provides high-precision simulation results that verify existing theoretical predictions in 2D and 4D, and offers a new estimate for the scaling exponent in 3D.

Findings

Confirmed the D=5/4 scaling in 2D.

Measured D=1.6236±0.0004 in 3D, differing from previous results.

Observed deviations from naive scaling in 4D consistent with two-loop predictions.

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