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

We derive the critical line of the O(n) loop model on the martini lattice as a function of the loop weight n basing on the critical points on the honeycomb lattice conjectured by Nienhuis [Phys. Rev. Lett. 49, 1062 (1982)]. In the limit n→0 we prove the connective constant μ=1.7505645579⋯ of self-avoiding walks on the martini lattice. A finite-size scaling analysis based on transfer matrix calculations is also performed. The numerical results coincide with the theoretical predictions with a very high accuracy. Using similar numerical methods, we also study the O(n) loop model on the 3-12 lattice. We obtain similarly precise agreement with the critical points given by Batchelor [J. Stat. Phys. 92, 1203 (1998)].