Polymer chain statistics and universality: Crossover from random to self-avoiding walks

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Abstract
We develop a position space renormalization group (PSRG) method to study the scaling properties of the Domb–Joyce model of an interacting random walk on a lattice. In this model each intersection of the walk with itself receives a weight (1−w). We study the crossover from unrestricted random walks (w=0) to self‐avoiding walks (w=1) using a two‐parameter PSRG approach, obtain the global flow diagram, and find the result consistent with universality, that the critical behavior for w>0 is described by the self‐avoiding walk fixed point. Our PSRG method avoids the difficulty of enumerating the infinite number of spanning random walks in a finite cell. The results for the mean end‐to‐end length exponent ν in spatial dimensiond=1, 2, and 3 are consistent with the exact result ν=1/2 for unrestricted random walks.

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