Regularity properties and pathologies of position-space renormalization-group transformations

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, de ned on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of de nition. This rules out a recently proposed scenario for the RG description of rst-order phase transitions. On the pathological side, we make rigorous some arguments of Grifths, 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-de ned, and that the conventional RG description of rst-order phase transitions is not universally valid. For decimation or Kadano transformations applied to the Ising model in dimension d 3, these pathologies occur in a full neighborhood f > 0 ; jhj < ( )g of the low-temperature part of the rstorder phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d 2, the pathologies occur at low temperatures for arbitrary magnetic-eld strength. Pathologies may also occur in the critical region for Ising models in dimension d 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.

Low-Temperature Dynamics of the Curie-Weiss Model: Periodic Orbits, Multiple Histories, and Loss of Gibbsianness

“…Historically they were observed first in the context of position-space renormalization group transformation and termed as so-called RG pathologies [11]. Later more and more examples were discovered [2,4,8,19,24] which showed that the application of many maps applied to an infinite-volume Gibbs measure may result in similar "pathologies", meaning that the image measure is not a Gibbs measure anymore. When such a phenomenon appears it means that conditional probabilities of the image system will acquire long-range dependencies, at least for some non-removable configurations.…”

Instability of renormalization-group pathologies under decimation

“…As the authors of ref. 4 point out, the main (rather surprising) pathology of the above RGT is that the renormalized measure v can very well be non-Gibbsian, that is, the associated system of conditional probabilities is not compatible with any finite-norm potential. That may happen even if the starting measure p, e.g., the unique Gibbs measure of some finite-range interaction, has all the nice properties describing the one-phase region: analyticity, exponential decay of correlations, convergent cluster expansion, etc.…”

“…In ref. 4 one can find many examples of such pathology. Moreover, the same authors show that the typical mechanism behind the non-Gibbsianness of the measure v is the appearance of long-range order, that is, a phase transition, in the object system conditioned to some particular configurations of the image system.…”

“…8 and Theorem2.12 in ref. 4), roughly speaking implies some sort of uniform continuity of the conditional probabilities rc A with respect to the conditioning configuration (see also refs. 1 and 5 for a critical discussion).…”

“…An interesting example given in ref. 4 of the above phenomenon refers to the "decimation transformation" on scale b, Tb a, applied to the Gibbs measure IZp.h of the Ising model at low temperature, fl>> 1, and small magnetic field h. Such a transformation associates to the original measure Pp.h its marginal (or relativization) on the spin variables sitting on the sites of the sublattice Zd(b) of Z d with spacing b. In other words, one integrates out all the variables in Zd\zd(b).…”