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

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 $\mu =\mathrm{1.7505645579...}$ 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)].