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 dimension d=1, 2, and 3 are consistent with the exact result ν=1/2 for unrestricted random walks.
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