Using Yang-Lee theory of phase transition and our extension, it is found that for a real fluid, both the singularity of canonical partition function and the critical point of the gas-liquid phase transition occur precisely at the temperature when all the cluster integrals become positive. The critical temperature is determined by the limit of the first zeros of the cluster integrals.

Based on Meakin' s simulation results [Phys. Rev. A 34, 710 (1986); 35, 2234 (1987)], we use the bino-mial distribution to discuss the growth probability for the screened-growth model and develop a formula of the generalized dimensions. The results from this model are in good agreement with those from the simulations.

We use the real-space renormalization-group approach to calculate the fractal dimension of diffusion-limited aggregation for a alttice with an arbitrary Euclidean dimension d. The results obtained are in good agreement with numerical numerical results. We prove that D → d-1 as d → ∞

We show that the singularity of the free energy of Ising models in the absence of a magnetic field on the triangular, square, and honeycomb lattices is related to zeros of the pseudopartition function on an elementary cycle. Using the Griffiths' smoothness postulate, we extend these results to the case in a magnetic field and derive a formula of the critical line of an Ising antiferromagnet, which is in good agreement with the numerical results.

Using the approach that we developed recently, we find the critical line of an anisotropic Ising antiferromagnet on two-dimensional square and honeycomb lattices. We extend our previous lemma and conjecture to be useful in the antiferromagnetic system. We find two interesting behaviors: (1) An antiferromagnet is not necessarily most inert at the absolute zero. (2) The field-driven antiferromagnetic phase transition is possible in the honeycomb lattice.