Exact critical points of the O($n$) loop model on the martini and the 3-12 lattices

TL;DR

This paper derives the exact critical line of the O(n) loop model on the martini and 3-12 lattices, confirming theoretical predictions with high-precision numerical methods and providing exact connective constants for self-avoiding walks.

Contribution

It presents the first exact critical line formula for the O(n) loop model on these lattices and validates it through highly accurate numerical analysis.

Findings

Exact critical line derived for the martini lattice.

High-precision numerical results match theoretical predictions.

Exact connective constant for self-avoiding walks on the martini lattice.

Abstract

We derive the exact critical line of the O($n$) loop model on the martini lattice as a function of the loop weight $n$.A finite-size scaling analysis based on transfer matrix calculations is also performed.The numerical results coincide with the theoretical predictions with an accuracy up to 9 decimal places. In the limit $n\to 0$, this gives the exact connective constant $\mu=1.7505645579...$ of self-avoiding walks on the martini lattice. Using similar numerical methods, we also study the O($n$) loop model on the 3-12 lattice. We obtain similarly precise agreement with the exact critical points given by Batchelor [J. Stat. Phys. 92, 1203 (1998)].