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

TL;DR

This study numerically investigates the critical exponents governing interchain correlations in self-avoiding walks and polygons, revealing a simple structure and generalizing previous exponents, with implications for understanding topological constraints.

Contribution

It introduces a numerical method to evaluate interchain correlation exponents for SAW and SAP, extending des Cloizeaux's exponents and analyzing their crossover behavior.

Findings

Exponents $ heta(i,j)$ have a simple, expressible structure.

Generalization of des Cloizeaux's three critical exponents.

Evaluation of diffusion coefficients for knotted SAPs.

Abstract

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 $\theta(i,j)$ 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 $\theta(i,j)$, and express it as a function of $i$ and $j$. We suggest that it has quite a simple structure. Here exponents $\theta(i,j)$ 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.