# Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps

## TL;DR

This study uses large-scale Monte Carlo simulations to analyze four-dimensional self-avoiding walks, confirming theoretical predictions about logarithmic corrections to scaling and providing insights into the efficiency of the pivot algorithm.

## Contribution

It provides the first extensive numerical evidence supporting the predicted logarithmic correction exponent for four-dimensional self-avoiding walks.

## Key findings

- Logarithmic correction exponent estimated at 0.2516(14).
- Pivot algorithm success probability scales as O([log N]^{-1/4}).
- Supports renormalization group predictions for 4D self-avoiding walks.

## Abstract

We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four-dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an $N$-step walk is $O([ \log N ]^{-1/4})$.