Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

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 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.Comment: 273 pages including 14 figures, Postscript, See also ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.

Vertical distribution of fish larvae in the Canaries-African coastal transition zone, in summer

13 pages, 6 figures, 2 tables.-- Printed version published Jul 2006.This study reports the vertical distribution of fish larvae during the 1999 summer upwelling season in the Canaries-African Coastal Transition Zone (the Canaries-ACTZ). The transition between the African coastal upwelling and the typical subtropical offshore conditions is a region of intense mesoscale activity that supports a larval fish population dominated by African neritic species. During the study, the thermal stratification extended almost to the surface everywhere, and the surface mixed layer was typically shallow or non-existent. Upwelling occurred on the African shelf in a limited coastal sub-area of our sampling. The vertical distributions of the entire larval fish population, as well as of individual species, were independent of the seasonal thermocline. Fish larvae and mesozooplankton were concentrated at intermediate depths regardless of the thermocline position, probably because of its weak signature and spatial and temporal variability. Day/night vertical distributions suggest that some species did not perform diel vertical migration (DVM), whereas others showed either type I DVM or type II DVM. The opposing DVM patterns of different species compensate for each other resulting in no net DVM for the larval fish population as a whole.Fieldwork was carried out as part of the CANIGO project, funded by the EU, and of the "Pelagic (EU-CICYT 1FD97-1084)" project from the Spanish Ministry of Education and the European Union