Three theorems in discrete random geometry
Probability Surveys 8 (2011) 403–441, arxiv
Theorem 2.2: The lower bound of d for the connective constant assumes a stronger notion of vertex-transitivity than implied by the conditions of the theorem. The lower bound is not satisfied by the hexagonal lattice.
The best such lower bound known to the author is Theorem 1.1 of Grimmett/Li.