The winding angle of planar self-avoiding walks

Abstract

The problem of understanding the asymptotic statistical behaviour of the winding angle, theta N, of a self-avoiding walk of N steps on a planar lattice is posed. Exact series expansion data for the square lattice up to N=21 are reported. These data and Monte Carlo estimates up to N<or approximately=170 steps are fitted well by a logarithmic growth law for ( theta N2). The ratio ( theta N4)/( theta N2)2 appears to approach a limit of 2.9 to 3.2, which is close to the Gaussian value, 3. Heuristic scaling arguments are consistent with simple logarithmic growth (and also illuminate the logarithmic behaviour known rigorously for free Brownian motion).