Growth of clusters and scale-free distribution at the phase transition point of the two-dimensional Q2R-Potts model

In this article, we present a study of the macroscopic and microscopic dynamics of the two-dimensional cellular automaton Q2R-Potts, which is an extension of the Q2R model. This extension takes into account interactions between more than two states. Specifically, we simulated its dynamics for three states (q=3). We observed that the system undergoes a second-order phase transition starting from a critical energy value E_{c}/N, which is equivalent to the critical value of the Potts model. Motivated by the observation of microscopic dynamics, we provide an exact characterization and quantification of cluster formation through subgraph connections. This allowed us to determine the critical cluster size that occurs during the phase transition, the development of a scale-free distribution, and the domain of states that form a chessboardlike structure.