Percolation Of The Site Random-cluster Model By Monte Carlo Method

Phase transition phenomena exist universally in nature, such as the transitions between gas, liquid and solid. Similarly, phase transition phenomena can be observed in statistical lattice models, for example, the Ising, Potts, XY and O(n) loop models. Apart from these statistical lattice models, there is another interesting model, i.e., random-cluster model. Universality of phase transitions unites the random-cluster model and these statistical lattice models together. Random-cluster model is classified into bond random-cluster model and site random-cluster model proposed in this thesis. The site random-cluster model is a statistical model by combining a cluster weight factorcnq with the partition function of the simple site percolation model in statistical physics. To verify whether or not the universality of the site random-cluster model and the bond random-cluster model is consistent, we design a high efficient Monte Carlo algorithm to simulate the site random-cluster model. Our research results are helpful for the understanding of the percolation of traditional statistical models.The main content of this thesis is arranged as follows:In the first chapter, we introduce the background knowledge of phase transition and percolation, and show the concept of fractal dimension and universality.In the second chapter, basing on the Ising, Potts and bond random-cluster models, we introduce the site random-cluster model and give its partition function format. Meanwhile, the finite-size scaling theory is introduced in detail.In the third chapter, we review the basic idea of Monte Carlo algorithm and introduce four kinds of Monte Carlo algorithms: Metropolis algorithm, Wolff algorithm, Swendsen-Wang algorithm, color-assignation algorithm. In addition,by combining color-assignation with Swendsen-Wang algorithm together, we design an efficient cluster algorithm to simulate the site random-cluster model;Finally, a method to judge whether or not the cluster is percolation is provided.In chapter IV, we apply the new cluster algorithm to simulate the site random-cluster model on the square lattice finding that for different exponents of cluster weight q=1.5, 2, 2.5, 3, 3.5 and 4, the numerical estimation of the exponents yt and yh 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 percolation strength P∞and the energy per site E.In chapter V, conclusion is presented.