Correction-to-scaling exponents for two-dimensional self-avoiding walks

TL;DR

This paper investigates correction-to-scaling exponents for 2D self-avoiding walks using enumeration and Monte Carlo methods, providing evidence for a specific non-analytic correction term and confirming field-theoretic predictions.

Contribution

It offers new precise estimates of correction-to-scaling exponents and confirms the value of the non-analytic correction term Delta_1 = 3/2 for 2D self-avoiding walks.

Findings

Evidence that the first non-analytic correction term is Delta_1 = 3/2

Good agreement with field-theoretic predictions for endpoint distribution moments

Cancellation of the leading analytic correction in a specific invariant ratio

Abstract

We study the correction-to-scaling exponents for the two-dimensional self-avoiding walk, using a combination of series-extrapolation and Monte Carlo methods. We enumerate all self-avoiding walks up to 59 steps on the square lattice, and up to 40 steps on the triangular lattice, measuring the mean-square end-to-end distance, the mean-square radius of gyration and the mean-square distance of a monomer from the endpoints. The complete endpoint distribution is also calculated for self-avoiding walks up to 32 steps (square) and up to 22 steps (triangular). We also generate self-avoiding walks on the square lattice by Monte Carlo, using the pivot algorithm, obtaining the mean-square radii to ~0.01% accuracy up to N = 4000. We give compelling evidence that the first non-analytic correction term for two-dimensional self-avoiding walks is Delta_1 = 3/2. We compute several moments of the endpoint distribution function, finding good agreement with the field-theoretic predictions. Finally, we study a particular invariant ratio that can be shown, by conformal-field-theory arguments, to vanish asymptotically, and we find the cancellation of the leading analytic correction.