 92142 A.C.D.van Enter , R.Fernandez , A.D.Sokal
 Regularity Properties and Pathologies
of PositionSpace RenormalizationGroup Transformations
(3779K, ps)
Oct 19, 92

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Abstract. We reconsider the conceptual foundations of the renormalizationgroup (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the
RG map, defined on a suitable space of interactions (= formal Hamiltonians),
is always singlevalued and Lipschitz continuous on its domain of definition.
This rules out a recently proposed scenario for the RG description of
firstorder phase transitions. On the pathological side, we make rigorous
some arguments of Griffiths, Pearce and Israel, and prove in several cases
that the renormalized measure is not a Gibbs measure for any reasonable
interaction. This means that the RG map is illdefined, and that the
conventional RG description of firstorder phase transitions is not
universally valid. For decimation or Kadanoff transformations applied to the
Ising model in dimension $d \ge 3$, these pathologies occur in a full
neighborhood $\{ \beta > \beta_0 ,\, h < \epsilon(\beta) \}$ of the
lowtemperature part of the firstorder phasetransition surface. For
blockaveraging transformations applied to the Ising model in dimension
$d \ge 2$, the pathologies occur at low temperatures for arbitrary
magneticfield strength. Pathologies may also occur in the critical region
for Ising models in dimension $d \ge 4$. We discuss in detail the
distinction between Gibbsian and nonGibbsian measures and the possible
occurrence of the latter in other situations, and give a rather complete
catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.
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