First-Order Theory of Self-Avoiding Walks Based on Loop Exclusion

Self-avoiding walks may be constructed through a progressive exclusion of walks with loops. A study of the process leads to critical exponents $\nu \simeq \frac{\mathrm{}(4+D)\mathrm{}}{4D}$ and $\gamma \simeq \frac{8}{\mathrm{}(4+D)\mathrm{}}$ for dimension $4>~D>~\frac{4}{3}$. The equations agree with the $\epsilon$ expansion to first order, fit the (known) values for $D=3,2\mathrm{}$, and also those (suggested) for $D<\mathrm{}2\mathrm{}$. The probability of an exclusion due to a loop of length $j$ appears to be asymptotically equal to $(\gamma -1)j^{-2\mathrm{}}$, for $4>D>\mathrm{}\frac{4}{3}$ ("strong universality").