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.

Rapid rates of sperm DNA damage after activation in tench (Tinca tinca: Teleostei, Cyprinidae) measured using a sperm chromatin dispersion test

Spermatozoal haplotypic DNA is prone to damage, leading to male fertility problems. So far, the assessment of sperm DNA breakage has been challenging because protamines render the nuclear chromatin highly compacted. Here, we report the application of a new test to quantify DNA fragmentation in spermatozoa of an externally fertilizing teleost fish. The sperm chromatin dispersion (SCD) test uses a species-specific lysing solution to generate controlled protein depletion that, followed by DNA-specific fluorescent labelling, allows an easy morphological discrimination between nuclei affected by DNA damage. Using tench (Tinca tinca) as our model, we first trialled the test against established, but more technically demanding, assays employing in situ nick translation (ISNT) and the comet assay. The SCD test showed high concordance with ISNT, comet assay measures and a chromatin-swelling test, confirming the application of this straightforward SCD technique to various aspects of reproductive biology. Second, we examined between-male variation in DNA damage, and measured changes through time following spermatozoal activation. Between-male variation in the basal levels of average DNA damage ranged from 0 to 20% of sperm showing damage, and all showed increases in DNA fragmentation through time (0–60 min). The rates of DNA damage increase are the fastest so far recorded in sperm for a living organism, and may relate to the external fertilization mode. Our findings have relevance for broodstock selection and optimizing IVF protocols routinely used in modern aquaculture