COHERENT-ANOMALY METHOD IN SELF-AVOIDING WALK PROBLEMS

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_1r in the asymptotic form C_nr ≃ β_lrλⁿ_1r 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 from the series of FORW approximants, C_nr ≃ br^gμ(1 + a/r)ⁿ, as the envelope curve C_n ≃ b(ae/g)^gμⁿn^g. Numerical results are given by C_n ≃ 1.424n^0.27884.1507ⁿ and C_n ≃ 1.179n^0.158710.005ⁿ 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.