The critical surface fugacity of self-avoiding walks on a rotated honeycomb lattice

TL;DR

This paper proves the conjectured critical surface fugacity for self-avoiding walks on a rotated honeycomb lattice, extending previous results on the standard honeycomb lattice and confirming longstanding conjectures.

Contribution

It establishes the critical surface fugacity for self-avoiding walks on a rotated honeycomb lattice, using methods similar to those in prior work on the standard lattice.

Findings

Confirmed the conjectured critical surface fugacity for the rotated lattice.

Extended the mathematical techniques used for the standard lattice to a rotated version.

Validated the universality of the phase transition behavior across lattice orientations.

Abstract

In a recent paper by Beaton et al, it was proved that a model of self-avoiding walks on the honeycomb lattice, interacting with an impenetrable surface, undergoes an adsorption phase transition when the surface fugacity is $1+\sqrt{2}$. Their proof used a generalisation of an identity obtained by Duminil-Copin and Smirnov, and confirmed a conjecture of Batchelor and Yung. We consider a similar model of self-avoiding walk adsorption on the honeycomb lattice, but with the lattice rotated by $\pi/2$. For this model there also exists a conjecture for the critical surface fugacity, made in 1998 by Batchelor, Bennett-Wood and Owczarek. Using similar methods to Beaton et al, we prove that this is indeed the critical fugacity.