Self-similarity in the classification of finite-size scaling functions for toroidal boundary conditions.

The conventional periodic boundary conditions in two dimensions are extended to general boundary conditions, prescribed by primitive vector pairs that may not coincide with the coordinate axes. This extension is shown to be unambiguously specified by the twisting scheme. Equivalent relations between different twist settings are constructed explicitly. The classification of finite-size scaling functions is discussed based on the equivalent relations. A self-similar pattern for distinct classes of finite-size scaling functions is shown to appear on the plane that parametrizes the toroidal geometry.

Numerical renormalization-group approach to a sandpile.

Multiple topplings and grain redistribution are two essential features in sandpile dynamics. A renormalization group (RG) approach incorporating these features is investigated. The full enumeration of all relaxations involving such an approach is difficult. Instead, we developed an efficient procedure to sample the relaxations. We applied this RG scheme to a square lattice and a triangular lattice. As shown by the fixed point analysis on a square lattice, the effect of multiple topplings leads the resultant height probabilities towards the exact solution while the effect of grain redistribution does not.

Master equation approach to folding kinetics of lattice polymers based on conformation networks.

Based on the master equation with the inherent structure of conformation network, the authors investigate some important issues in the folding kinetics of lattice polymers. First, the topologies of conformation networks are discussed. Moreover, a new scheme of implementing Metropolis algorithm, which fulfills the condition of detailed balance, is proposed. Then, upon incorporating this new scheme into the geometric structure of conformation network the authors provide a theorem which can be used to place an upper bound on relaxation time. To effectively identify the kinetic traps of folding, the authors also introduce a new quantity, which is employed from the continuous time Monte Carlo method, called rigidity factor. Throughout the discussions, the authors analyze the results for different move sets to demonstrate the methods and to study the features of the kinetics of folding.

Partition functions and finite-size scalings of Ising model on helical tori.

The exact closed forms of the partition functions of a two-dimensional Ising model on square lattices with twisted boundary conditions are given. The constructions of helical tori are unambiguously related to the twisted boundary conditions by virtue of the SL(2, Z) transforms. The numerical analyses on the deviations of the specific-heat peaks away from the bulk critical temperature reveal that the finite-size effect of herical tori is independent of the chirality.

Evolution and structure formation of the distribution of partition function zeros: triangular type Ising lattices with cell decoration.

The distribution of partition function zeros of the two-dimensional Ising model in the complex temperature plane is studied within the context of triangular decorated lattices and their triangle-star transformations. Exact recursion relations for the zeros are deduced for the description of the evolution of the distribution of the zeros subject to the change of decoration level. In the limit of infinite decoration level, the decorated lattices essentially possess the Sierpinski gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpinski gasket or its triangle-star transformation, and the distributions of zeros all appear to be a union of infinite scattered points and a Jordan curve, which is the limit of the scattered points.