Two-dimensional self-avoiding walks on a cylinder

We present simulations of self-avoiding random walks (SAWs) on two-dimensional lattices with the topology of an infinitely long cylinder, in the limit where the cylinder circumference $L$ is much smaller than the Flory radius. We study in particular the $L$ dependence of the size $h$ parallel to the cylinder axis, the connectivity constant \ensuremath{\mu}, the variance of the winding number around the cylinder, and the density of parallel contacts. While $\ensuremath{\mu}(L)$ and $〈{W}^{2}(L,h)〉$ scale as expected [in particular, $〈{W}^{2}(L,h)〉\ensuremath{\sim}h/L],$ the number of parallel contacts decays as ${h/L}^{1.92},$ in striking contrast to recent predictions. These findings strongly speak against recent speculations that the critical exponent \ensuremath{\gamma} of SAWs might be nonuniversal. Finally, we find that the amplitude for $〈{W}^{2}〉$ does not agree with naive expectations from conformal invariance.