Two-dimensional self-avoiding walks and polymer adsorption: Critical fugacity estimates

TL;DR

This paper estimates the critical fugacity for self-avoiding walks interacting with surfaces on square and triangular lattices, extending known results from the honeycomb lattice and achieving high precision in these estimates.

Contribution

It provides the first accurate estimates of critical fugacity for square and triangular lattices, and proves the result for the honeycomb lattice with edge weighting.

Findings

Critical fugacity estimates for square and triangular lattices.

Proof of critical fugacity for honeycomb lattice with edge weighting.

High-precision numerical results for polymer-surface interactions.

Abstract

Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and Yung that the critical fugacity of self-avoiding walks interacting with (alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A key identity used in that proof depends on the existence of a parafermionic observable for self-avoiding walks interacting with a surface on the honeycomb lattice. Despite the absence of a corresponding observable for SAW on the square and triangular lattices, we show that in the limit of large lattices, some of the consequences observed for the honeycomb lattice persist irrespective of lattice. This permits the accurate estimation of the critical fugacity for the corresponding problem for the square and triangular lattices. We consider both edge and site weighting, and results of unprecedented precision are achieved. We also \emph{prove} the corresponding result fo the edge-weighted case for the honeycomb lattice.