A.C.D.van Enter , R.Fernandez , A.D.Sokal
Regularity Properties and Pathologies
of Position-Space Renormalization-Group Transformations
(3779K, ps)
ABSTRACT. We reconsider the conceptual foundations of the renormalization-group (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 single-valued and Lipschitz continuous on its domain of definition.
This rules out a recently proposed scenario for the RG description of
first-order 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 ill-defined, and that the
conventional RG description of first-order 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
low-temperature part of the first-order phase-transition surface. For
block-averaging transformations applied to the Ising model in dimension
$d \ge 2$, the pathologies occur at low temperatures for arbitrary
magnetic-field 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 non-Gibbsian 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.