Summary
We consider the Kadanoff transformation T (depending on a positive parameter p) acting on probability measures Μ on the space {+1, −}ℤd. A measure Μ is called a non-trivial fixed point of T, if it is extremal in the set of T-invariant measures but is not a product measure. We describe the set of trivial fixed points and show that non-trivial fixed points exist provided that d≦2 and p large enough. A strong mixing condition on Μ implies convergence of T nΜ towards a trivial fixed point. In particular this applies to the two-dimensional Ising model except at the critical point. What happens at the critical point still remains unknown.
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Research supported by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 123)
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Higuchi, Y., Lang, R. On the convergence of the kadanoff transformation towards trivial fixed points. Z. Wahrscheinlichkeitstheorie verw Gebiete 58, 109–123 (1981). https://doi.org/10.1007/BF00536199
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DOI: https://doi.org/10.1007/BF00536199