Percolation of the Site Random-Cluster Model by Monte Carlo Method

TL;DR

This paper introduces a novel site random cluster model, develops an efficient Monte Carlo simulation method, and demonstrates its consistency with the bond model, revealing phase transition behaviors and universal properties.

Contribution

It proposes a new site random cluster model with an efficient cluster algorithm, and verifies its properties and universality with the bond model through numerical simulations.

Findings

The model's critical exponents match theoretical values.

Universalities of site and bond models are identical.

First-order transition signatures appear for larger q values.

Abstract

Herein, we propose a site random cluster model by introducing an additional cluster weight in the partition function of the traditional site percolation. To simulate the model on a square lattice, we combine the color-assignation and the Swendsen-Wang methods together to design a highly efficient cluster algorithm with a small critical slowing-down phenomenon. To verify whether or not it is consistent with the bond random cluster model, we measure several quantities such as the wrapping probability $R_e$, the percolation strength $P_\infty$, and the magnetic susceptibility per site $\chi_p$ as well as two exponents such as the thermal exponent $y_t$ and the fractal dimension $y_h$ of the largest cluster. We find that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5 and 4, the numerical estimation of the exponents $y_t$ and $y_h$ are consistent with the theoretical values. The universalities of the site random cluster model and the bond random cluster model are completely identical. For larger values of $q$, we find obvious signatures of the first-order percolation transition by the histograms and the hysteresis loops of the percolation strength and the energy per site. Our results are helpful for the understanding of the percolation of traditional statistical models.