Finite size scaling in the dimer and six-vertex model

TL;DR

This paper compares Monte Carlo algorithms for finite size scaling in dimer and six-vertex models, confirming Gaussian free field behavior and highlighting the Wang-Landau algorithm's precision.

Contribution

It introduces a comparison of Markov-chain and Wang-Landau algorithms for dimer models, demonstrating the latter's superior accuracy in finite size scaling analysis.

Findings

Wang-Landau algorithm yields more precise results than Metropolis.

Logarithmic correlation function confirms Gaussian free field behavior.

Numerical results for non-hexagonal domains where theory is unavailable.

Abstract

We present results of the Monte-Carlo simulations for scaling of the free energy in dimers on the hexagonal lattice. The traditional Markov-chain Metropolis algorithm and more novel non-Markov Wang-Landau algorithm are applied. We compare the calculated results with the theoretical prediction for the equilateral hexagon and show that the latter algorithm gives more precise results for the dimer model. For a non-hexagonal domain the theoretical results are not available, so we present the numerical results for a certain geometry of the domain. We also study the two-point correlation function in simulations of dimers and the six-vertex model. The logarithmic dependence of the correlation function on the distance, which is in accordance with the Gaussian free field description of fluctuations, is obtained.