Journal of Statistical Physics, Vol. 86. Nos. 5/6. 1997. p. 1089-1107.
We consider the majority-rule renormalization group translbrmation applied to nearest neighbor lsing models. For the square lattice with 2 by 2 blocks we prove that if the temperature is sufficiently low, then the translbrmation is not delined. We use the methods of van Enter, Fernandez, and Sokal, who proved the renormalized measure is not Gibbsian for 7 by 7 blocks if the temperature is too low. For the triangular lattice we prove that a zero-temperature majority-rule translbrmation may be defined. The resulting renormalized Hamiltonian is local with 14 different types of interactions.