Coherent-anomaly method in self-avoiding walk problems

Abstract

Self-avoiding walk (SAW), being a nonequilibrium cooperative phenomenon, is investigated with a finite-order-restricted-walk (finite-ORW or FORW) coherent-anomaly method (CAM). The coefficient β 1 r in the asymptotic form C nr ≅ β 1 r λ n 1 r for the total number C nr of r- ORW's with respect to the step number n is investigated for the first time. An asymptotic form for SAW's is thus obtained form the series of FORW approximants, C nr ≅ br gμ n(1 + a/r) n , as the envelope curve C n ≅b(ae/g) gμ nn g. Numerical results are given by C n≅1.424 n 0.27884.1507 n and C n ≅1.179 n 0.158710.005 n for the plane triangular lattice and f.c.c. lattice, respectively. A good coincidence of the total numbers estimated from the above simple formulae with exact enumerations for finite-step SAW's implies that the essential nature of SAW (non-Markov process) can be understood from FORW (Markov process) in the CAM framework.