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

We show numerically that critical exponents for two-point interchain 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 $i$th and $j$th vertices of $N$-step SAW (or SAP with a knot) for all pairs ($1 \le i, j \le N$). 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 interchain 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.

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