Fractal properties of polymers on lattices

The square and cubic lattice percolation problem and the self-avoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and linear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand.. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks.