New spin model for the biased self-avoiding walk

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Abstract
We present a new spin model for the biased self-avoiding walk on the square lattice. Unlike a previous model containing both vector and Ising spins, our model contains only n-vector spins, evaluated in the n→0 limit, and the spin Hamiltonian is Hermitian. Integration over one of the three sets of spins leads to a rigorous proof that the crossover exponent from straight chain to random walk behavior is 1. An analysis of the scaling behavior of the Hamiltonian in the small gauche-bond probability limit enables us to conclude that self-interaction effects in the biased walk will be irrelevant in three dimensions, but not in two dimensions, in agreement with some earlier numerical results.

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