Abstract
A method for computing low--temperature series for renormalized operators in the two--dimensional Ising model is proposed. These series are applied to the study of the properties of the truncated renormalized Hamiltonians when we start at very low temperature and zero field. The truncated Hamiltonians for majority rule, Kadanoff transformation and decimation for $2 \times 2$ blocks depend on the how we approach the first--order phase--transition line. These Renormalization Group transformations are multi--valued and discontinuous at this first--order transition line when restricted to some finite--dimensional interaction space.
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