REGULARITY PROPERTIES AND PATHOLOGIES OF POSITION-SPACE RENORMALIZATION-GROUP TRANSFORMATIONS - SCOPE AND LIMITATIONS OF GIBBSIAN THEORY

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.