Particle density and critical point for studying site percolation by finite size scaling

Machine learning has recently achieved remarkable success in studying phase transitions. It is generally believed that the latent variables of unsupervised learning can capture the information related to phase transitions, which is usually achieved through the so-called order parameter. In most models, for instance the Ising, the order parameters are simply the particle number densities. The percolation, the simplest model which can generate a phase transition, however, has a unique order parameter which is not particle number density. In this paper, we use unsupervised learning to study the relationship between particle number density, critical point, and latent variables in the site percolation model. It is found that if the input of learning is the original configuration, then the output of unsupervised learning does not convey any information related to the phase transition. Therefore, the maximum cluster is employed in order to effectively capture the critical point of the model. Unsupervised learning yields reliable results consistent with Monte Carlo simulations. We also propose a method called Fake Finite Size Scaling (FFSS) to calculate the critical value, which improves the accuracy of fitting to a great extent.