Exponents of intrachain correlation for self-avoiding walks and knotted self-avoiding polygons

Abstract

We show numerically that critical exponents for two-point intrachain correlation of an infinite chain characterize those of finite chains in self-avoiding walk (SAW) and self-avoiding polygon (SAP) under a topological constraint. We evaluate short-distance exponents θ(i, j) through the probability distribution functions of the distance between the ith and jth vertices of N-step SAW (or SAP with a knot) for all pairs (1 ⩽ i, j ⩽ N). We construct the contour plot of θ(i, j), and express it as a function of i and j. We suggest that it has quite a simple structure. Here exponents θ(i, j) generalize des Cloizeaux’s three critical exponents for short-distance intrachain correlation of SAW, and we show the crossover among them. We also evaluate the diffusion coefficient of knotted SAP for a few knot types, which can be calculated with the probability distribution functions of the distance between two nodes.