Study of Phase Transitions and Universality Classes in Percolation and Ising Models

Abstract

This dissertation investigates phase transitions and universality classes within percolation and Ising models, employing computational simulation methods. For percolation models, the study demonstrates that lattice disorder does not alter the universality class, while site occupation incorporating continuous random walks can induce changes in the universality class, depending on the step length. Additionally, the research constructs a set of percolation phase transitions sharing the same universality class within fractal graphs derived from the Sierpiński carpet. In all cases, the type of phase transition remains consistent with that of the standard percolation model. In the case of Ising models, the study explores the J1-J2 Ising model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions on two generalized triangular lattices. Depending on the ratio of J2 to J1, the type of phase transition changes in anisotropic lattices, whereas two successive phase transitions occur in non-regular lattices. Furthermore, the thermodynamic and dynamic phase transitions of the Ising model on fractal graphs based on the Sierpiński carpet are examined, verifying that these transitions belong to the same universality class.