Asymptotic behavior of the "true" self-avoiding walk

The "true" self-avoiding random walk is defined as the statistical problem of a traveler who steps randomly, but tries to avoid places he has already visited. We show that this problem is different from the problem of a self-repelling chain (polymer problem). Most striking is perhaps the fact that the upper critical dimensionality of such a walk is 2. Renormalization-group theory is applied to compute logarithmic corrections to ordinary random-walk behavior in two dimensions. The theoretical predictions are confirmed by computer simulations.