Theta-point polymers in the plane and Schramm-Loewner evolution

TL;DR

This paper establishes a connection between lattice polymers at the theta temperature and Schramm-Loewner chains, demonstrating convergence of the driving function to Brownian motion and matching critical exponents with theoretical predictions.

Contribution

It introduces a continuum Schramm-Loewner chain approach to sample tricritical polymers at the theta point, linking lattice models with conformal invariance.

Findings

Driving function converges to Brownian motion with kappa=6

Correlation length exponent nu=4/7 matches predictions

Shape factor and asphericity agree with theta-point values

Abstract

We study the connection between polymers at the theta temperature on the lattice and Schramm-Loewner chains with constant step length in the continuum. The latter realize a useful algorithm for the exact sampling of tricritical polymers, where finite-chain effects are excluded. The driving function computed from the lattice model via a radial implementation of the zipper method is shown to converge to Brownian motion of diffusivity kappa=6 for large times. The distribution function of an internal portion of walk is well approximated by that obtained from Schramm-Loewner chains. The exponent of the correlation length nu and the leading correction-to scaling exponent Delta_1 measured in the continuum are compatible with nu=4/7 (predicted for the theta point) and Delta_1=72/91 (predicted for percolation). Finally, we compute the shape factor and the asphericity of the chains, finding surprising accord with the theta-point end-to-end values.