A catalytic reaction model, the Ziff-Gulari-Barshad model, is studied on fractal lattices, and the influence of the order of ramification of the lattice on the dynamic behavior of the model is investigated. According to the Monte Carlo simulation results, the order of ramification of the lattice is not crucial to the existence of the continuous transition. This is different from the equilibrium phase transitions in discrete-symmetry spin models such as the Ising model!. Our results indicate that the criterion of the existence of the reactive phase may be complicated.

The Potts antiferromagnet situated on the fractal family of diamond-type hierarchical lattices is investigated The exact analysis of our findings reveals that the algebraically ordered behavior pre- dicted by Berker and Kadanoff can exist on such lattices with L=odd, where L is the structure pa- rameter of the hierarchical lattices; however, similar behavior to that of the antiferromagnetic Potts model on a decorated square lattice is exhibited for systems with L=even. We find that the different constructions that have the same ffactal dimensionality can have a different cutoff value q 0, in contrast with the translationally invariant system. We also give the ground-state entropy at q=qo in a closed form for the systems with L=even.

The critical dynamics of the kinetic Glauber-Ising model is studied on a family of the diamond-type hierarchical lattices with various branches. By carrying out the time-dependent real-space renormalizationgroup transformation to the master equation of the systems considered, the dynamic exponent is calculated. We find that the dynamic exponent depends on fractal dimension df or the branch number m in a generator, and that it increases with the increase of df or m. We notice that for the case of m51 one-dimensional spin chain, df51) our result z52 is the same as the exact result obtained by Glauber, and for the case of m52 the simplest one in the diamond-type hierarchical lattices, df52) the exponent z52.626 is higher than those of the two-dimensional regular lattice and the triangular lattice.

In this paper, we present a multispin transition mechanism, which is an extension of the Glauber one, to investigate critical dynamics. By exactly solving the master equation, the influence of the multispin transition mechanism on the dynamic critical behavior is studied for the Gaussian model with nearest-neighbor interactions on d-dimensional lattices (d51, 2, and 3!. The time evolution of magnetization is exactly calculated, and the exact results of relaxation time and dynamic critical exponent are obtained. Our models are divided into two kinds: one is the spin-cluster transition and the other is the arbitrary multispin transition. It is found that there are different relaxation times, but the same dynamical critical exponent for different kinds of multispin transitions. The results show that the dynamical critical exponents are independent of spatial dimensions and configurations of transitional spins, and that the dynamical critical exponent is the same as that of the Glauber dynamics, and thus give a strong support to the simple single-spin-transition dynamics. Finally, we give a brief discussion on the results.

The partition function of the lsing model on the Sierpinski gasket is obtained by using a decoupled Sierpinski gasket Ising model as an unperturbed system. The exact partition function can be expressed as a statistical average of a special product over an unperturbed Ising Hamiltonian. A high-temperature expansion of the partition function is produced.