Growing self avoiding walk trees

The Growing self avoiding walk model (GSAW) was proposed to explain statistical mechanics of the growth process in polymerization. Close examination of the model reveals that it is suited only for addition polymerization where only one monomer unit is added to the growing polymer chain. In this paper we propose a model for step growth or condensation polymerisation, where all of the monomer is converted first to dimer, all dimers are converted to tetramer etc. In the calculation of probabilities of the walker taking a step in a specified direction in the GSAW model the probability is the reciprocal of the number of steps available if the walker looks one step ahead. It is seen that our model for condensation polymerization GSAW can be generalized to the walker looking a finite number of steps ahead (instead of one only) for the purpose of calculation of probabilities. This is explained in detail in the first section of the paper. In the second section of the paper we explain the use of the Depth first search (DFS) algorithm in the calculation of probabilities and moments for the generalized growing self avoiding walk model. In the third section of the paper we report the exact values of the mean square end to end distances and relevant survival probabilities for the model. When GSAW was first proposed there was a controversy over whether it belonged to the same universality class as the True self avoiding walk \(\left( \upsilon =\frac{2}{d+1}\right) \) or that of the self avoiding walk \(\left( \upsilon =\frac{3}{d+2}\right) \). At that time it was conclusively shown that the GSAW model cannot belong to the same universality class as the true self avoiding walk model. However it was never conclusively shown that \(\upsilon =\frac{3}{d+2}\) for GSAW. In this paper we propose two methods of studying this problem. One method is rigorous analysis of the DFS algorithm. We explain how this algorithm can be used to study restricted random walks with finite memory and self avoiding walks. The second method proposed by us is a detailed analysis of the generalised GSAW model proposed by us. This paper is to be viewed as an introductory paper on a new model that should be of interest to both chemists and mathematicians.