Asymptotic properties of spiral self - avoiding walks

We consider the spiral self-avoiding walk model recently introduced by Privman. On the basis of an analogy with the partitioning of the integers, we argue that the number of N-step spiral walks should increase asymptotically as pJN with p a constant, leading to an essential singularity in the generating function. Enumeration data to 65 terms indicates, however, that c , apparently varies as pNn, with a -0.55. We also study the N-dependence of the mean-square end-to-end distance, (It:), and of the mean rotation angle, (ON), for N-step walks. From series extrapolations, we estimate that ( R L ) N ' *, and (e, ) Very recently, Privman (1983) introduced for the square lattice a 'spiral' self-avoiding walk (SAW) model, defined to be a SAW with the additional constraint that a 90" left-hand turn is not allowed. Accordingly, a walk will tend to spiral around the origin in the clockwise sense (figure 1) . On the basis of exact enumeration data to order 40, Privman argued that the spiral SAW model is in a different universality class than the isotropic SAW problem. This result was interpreted as arising from the fact that the spiral restriction is effectively global in nature when it augments the self-avoiding constraint. Well known examples of a global constraint modifying the universality class occur in 'directed' problems (see e.g. Kinzel 1983). The directionality constraint can be thought of as analogous to a uniform drift superimposed on a transport problem. This "3 :Inb