Acd. Vanenter et al., REGULARITY PROPERTIES AND PATHOLOGIES OF POSITION-SPACE RENORMALIZATION-GROUP TRANSFORMATIONS - SCOPE AND LIMITATIONS OF GIBBSIAN THEORY, Journal of statistical physics, 72(5-6), 1993, pp. 879-1167
We reconsider the conceptual foundations of the renormalization-group
(RG) formalism, and prove some rigorous theorems on the regularity pro
perties and possible pathologies of the RG map. Our main results apply
to local (in position space) RG maps acting on systems of bounded spi
ns (compact single-spin space). Regarding regularity, we show that the
RG map, defined on a suitable space of interactions (=formal Hamilton
ians), 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, an
d 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 p
hase transitions is not universally valid. For decimation or Kadanoff
transformations applied to the Ising model in dimension d greater-than
-or-equal-to 3, these pathologies occur in a full neighborhood {beta >
beta0, Absolute value of h < epsilon(beta)} of the low-temperature pa
rt of the first-order phase-transition surface. For block-averaging tr
ansformations applied to the Ising model in dimension d greater-than-o
r-equal-to 2, the pathologies occur at low temperatures for arbitrary
magnetic field strength. Pathologies may also occur in the critical re
gion for Ising models in dimension d greater-than-or-equal-to 4. We di
scuss the heuristic and numerical evidence on RG pathologies in the li
ght of our rigorous theorems. In addition, we discuss critically the c
oncept of Gibbs measure, which is at the heart of present-day classica
l statistical mechanics. We provide a careful, and, we hope, pedagogic
al, overview of the theory of Gibbsian measures as well as (the less f
amiliar) non-Gibbsian measures, emphasizing the distinction between th
ese two objects and the possible occurrence of the latter in different
physical situations. We give a rather complete catalogue of the known
examples of such occurrences. The main message of this paper is that,
despite a well-established tradition, Gibbsianness should not be take
n for granted.