IRREVERSIBLE SELF AVOIDING WALKS

Abstract

This chapter discusses two new self avoiding walks (SAW) that possess the property of irreversibility, that is, the product of the one step probabilities is different in the two directions along the chain. The first walk, which is called the growing SAW (GSAW), can be viewed as an extension of the recently introduced true SAW. In the infinite interaction energy limit, this walk checks if it has visited nearest neighbors already and then travels to a site that has not been visited before with probability. A difficulty occurs when all nearest neighbor sites are occupied. The true SAW proceeds with a probability of 1 / (# n.n. sites) to an already visited site. The chapter introduces the self-avoiding property, that is, when a walk locks itself up in a cage, it is terminated as it is done for the static SAW when it tries to visit a site for the second time. This walk could serve as a model to study the growth process of a linear polymer in a “good― solvent as long as the relaxation of the chain is slower than the growth procedure. The second walk (Smart GSAW) has been constructed in such a way that it avoids cages. These two walks were studied using enumeration and real space renormalization techniques. The GSAW was studied in two dimensions on the square lattice and the triangular lattice and in three dimensions on the diamond lattice. The enumeration results were obtained by exactly counting all walks.