The O(n) loop model on the 3-12 lattice

TL;DR

This paper establishes a mapping between the O(n) loop model on the honeycomb and 3-12 lattices, revealing their shared critical behavior and providing exact critical points for specific cases like self-avoiding walks and the Ising model.

Contribution

It introduces a mapping that relates the O(n) loop models on two different lattices, enabling the derivation of critical points and exponents for these models.

Findings

Critical points are related via a simple variable transformation.

Exact connective constant for self-avoiding walks on the 3-12 lattice is obtained.

Exact critical points for the Ising model on the 3-12 and dual asanoha lattices are recovered.

Abstract

The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3-12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n=0 this gives the recently found exact value $\mu = 1.711 041...$ for the connective constant of self-avoiding walks on the 3-12 lattice. The exact critical points are recovered for the Ising model on the 3-12 lattice and the dual asanoha lattice at n=1.