Lagrangian theory for a self-avoiding random chain

Abstract

The Lagrangian theory of random chains with excluded volume is used to study $Z_N\left(\overrightarrow{\mathrm{r}}\right)$, the number of chains with $N$ links, starting from the origin and arriving at a point $\overrightarrow{\mathrm{r}}$. Its asymptotic expression ($N\to \infty$) is $Z_N\mathrm{}\left(r\right)\mathrm{}\simeq N^{\gamma -1-\nu d} F\left(r{N}^{-\nu }\right)$, where $\gamma$ and $\nu$ are critical indices. The short- and long-range behaviors of $F\left(x\right)\mathrm{}$ are calculated in terms of $\gamma$ and $\nu$. In particular, it is shown that for $x\ll 1$, we have $F\left(x\right)\mathrm{}\simeq F{x}^{\theta }$ with $F=\mathrm{const}$ and $\theta =\frac{(\gamma -1)}{\nu }$.