Equilibrium winding angle of a polymer around a bar

The winding angle probability distribution of a planar self-avoiding walk has been known exactly since a long time: it has a gaussian shape with a variance growing as $<{\theta }^2>\sim \ln L$. For the three-dimensional case of a walk winding around a bar, the same scaling is suggested, based on a first-order epsilon-expansion. We tested this three-dimensional case by means of Monte Carlo simulations up to length $L\approx 25000$ and using exact enumeration data for sizes $L\le 20$. We find that the variance of the winding angle scales as $<{\theta }^2>\sim (\ln L{)}^{2\alpha }$, with $\alpha =0.75(1)$. The ratio $\gamma = <\theta^4>/<\theta^2>^2=3.74(5)$isincompatiblewiththegaussianvalue$\gamma =3$, but consistent with the observation that the tail of the probability distribution function $p(\theta )$ is found to decrease slower than a gaussian function. These findings are at odds with the existing first-order $ϵ$-expansion results.