The transition matrix approach to selfavoiding walks

A restricted walk of order r is defined as a random walk in which polygons with r vertices or less are excluded. The sequence of restricted walks approaches a selfavoiding walk in the limit r to infinity . Although exact enumerations and Monte Carlo runs provide better data for numerical analysis, a study of restricted walks for increasing r provides an understanding of how the transition in properties is effected from random to selfavoiding walks. The eigenvalues of the transition matrix governing a restricted walk have been calculated up to r=7 for the triangular lattice and r=5 for the fcc lattice. It is found that the dominant contribution to the total number of walks comes from the largest eigenvalue lambda 1r which satisfies approximately the relation lambda 1r approximately= mu (1+g/r). Here mu is the selfavoiding walk limit and g=1/3 for the triangular and 1/6 for the fcc lattice.