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The critical behaviour of s-dimensional fully directed flights (FDFs) is treated by a generating function method and an exact calculation method. Our results show that the exponents ν, ν∥ and ν⊥ of FDFs are independent of the spatial dimensionality and the concrete form of the probability distribution of step lengths. They are in agreement with the Laplacian transformation method and the Monte Carlo simulation study for fully directed Levy flights.

The critical behaviour of conductivity for some two-dimensional fractals is studied by experiments. The results show that the conductivity exponent is fractal dimensionality dependent with exponential scaling behaviour but is fractal material independent.

The atomic aggregation processes in the early stages of Pt/Pt(111) growth are studied by using kinetic Monte Carlo simulations. Our results show that the average neighbor coordination number of the atoms in a cluster is a function of temperature, agreeing well with the experiment observations of scanning tunneling microscopy. The influence of diffusion barriers of various atomic processes on the morphology of islands is also studied. Different morphologies of the islands (dendritic, fractal, or compact islands) are found.

The deposition, diffusion and coalescence processes appeared in the surface growth of compact or liquid-like clusters system are studied. By introducing a critical cluster size ic, such that clusters of size s ≤ ic are mobile while clusters of the size s > ic are immobile, the effects of mobility of small clusters on cluster size distribution are simulated. As ic increases, the exponent γ which relates the maximum cluster density to the normalized deposition rate is increased. Numerical simulation results indicate that the scaled cluster-size distributions are independent of the deposition rate, but closely associated with the small clusters mobility. As ic increases, it is found that the distributions become narrow.

Using the fractal-cell generation method we perform a numerical simulation study for partially directed self-avoiding walks (PDSAW) on Sierpinski carpets. The obtained critical exponents is found to be independent of the fractal dimension of Sierpinski carpet , but is dependent on . This result indicates that PDSAW on different Sierpinski carpets belong to different universality classes. Compared with the fully directed self-avoiding walks (FDSAW) on the same carpets, the obtained results indicate that PDSAW and FDSAW belong to the same universality class.

Levy flight is a random walk in which the step length r is a random variable with probability distribution proportional to 1/r1+u, where u is the Levy parameter and 0

The critical behaviour of fully directed self-avoiding walks (DSAW) on Sierpinski carpets is studied by numerical simulation. The authors propose a fractal cell generation method to form an infinite fractal lattice with the structure of a Sierpinski carpet. The obtained critical exponent ν|| is independent of the fractal dimension of a Sierpinski carpet df but ν and ν⊥ are dependent on df. The results indicate that DSAW on different Sierpinski carpets belong to different universality classes.