Scaling behaviour of lattice animals at the upper critical dimension

TL;DR

This study investigates the scaling behavior and logarithmic corrections of lattice animals at the upper critical dimension d=8 using advanced simulation techniques, confirming theoretical predictions and proposing new scaling relations.

Contribution

The paper provides high-precision numerical estimates of logarithmic corrections for lattice animals at the upper critical dimension and introduces a new scaling relation for these corrections.

Findings

Confirmed Parisi-Sourlas prediction for leading exponents

Estimated logarithmic correction exponents accurately

Proposed a new scaling relation for logarithmic corrections

Abstract

We perform numerical simulations of the lattice-animal problem at the upper critical dimension d=8 on hypercubic lattices in order to investigate logarithmic corrections to scaling there. Our stochastic sampling method is based on the pruned-enriched Rosenbluth method (PERM), appropriate to linear polymers, and yields high statistics with animals comprised of up to 8000 sites. We estimate both the partition sums (number of different animals) and the radii of gyration. We re-verify the Parisi-Sourlas prediction for the leading exponents and compare the logarithmic-correction exponents to two partially differing sets of predictions from the literature. Finally, we propose, and test, a new Parisi-Sourlas-type scaling relation appropriate for the logarithmic-correction exponents.