Persistency of two-dimensional self-avoiding walks

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Abstract
The authors investigate the persistency of self-avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step. Exact enumeration and the constant-fugacity Monte Carlo data indicate that the mean displacement after N steps (xN), increases logarithmically with N, while the higher odd moments (x2k+1N) vary as N2kv In N, where v is the correlation length exponent. This unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution, which in the constant-fugacity ensemble can be written in the form A(x) approximately N-2v In N exp(-bx/Nv), where N is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity.

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