Accurate Checks of Universality for Dyson's Hierarchical Model

Using recently developed methods, we perform high-accuracy calculations of the susceptibility near beta_c for the D=3 version of Dyson's hierarchical model. Using linear fits, we estimate the leading gamma and subleading Delta exponents. Independent estimates are obtained by calculating the first two eigenvalues of the linearized renormalization group transformation. We found gamma = 1.29914073 (with an estimated error of 10^{-8}) and, Delta=0.4259469 (with an estimated error of 10^{-7}) independently of the choice of local integration measure (Ising or Landau-Ginzburg). After a suitable rescaling, the approximate fixed points for a large class of local measure coincide accurately with a fixed point constructed by Koch and Wittwer.Comment: 9 pages, Revtex, 1 figur

High-accuracy critical exponents of O(N) hierarchical sigma models

We perform high-accuracy calculations of the critical exponent gamma and its subleading exponent for the 3D O(N) Dyson's hierarchical model, for N up to 20. We calculate the critical temperatures for the nonlinear sigma model measure. We discuss the possibility of extracting the first coefficients of the 1/N expansion from our numerical data. We show that the leading and subleading exponents agreewith Polchinski equation and the equivalent Litim equation, in the local potential approximation, with at least 4 significant digits.Comment: 4 pages, 2 Figs., uses revte

Symmetry Breaking, Duality and Fine-Tuning in Hierarchical Spin Models

We discuss three questions related to the critical behavior of hierarchical spin models: 1) the hyperscaling relations in the broken symmetry phase; 2) the combined use of dual expansions to calculate the non-universal quantities; 3) the fine-tuning issue in approximately supersymmetric models.Comment: 3 pages, 1 figure, Lattice99 (spin

Accounting for a spatial trend in fine-scale ground-penetrating radar data: A comparative case study

In geostatistics, one of the challenges is to account for the spatial trend that is evident in a data-set. Two well-known kriging algorithms, namely universal kriging (UK) and intrinsic random function of order k (IRF-k), are mainly used to deal with the trend apparent in the data-set. These two algorithms differ in the way they account for the trend and they both have different advantages and drawbacks. In this study, the performances of UK, IRF-k, and ordinary kriging (OK) methods are compared on densely sampled ground-penetrating radar (GPR) data acquired to assist in delineation of the ore and waste contact within a laterite-type bauxite deposit. The original GPR data was first pre-processed to generate prediction and validation data sets in order to compare the estimation performance of each kriging algorithm. The structural analysis required for each algorithm was carried out and the resulting variograms and generalized covariance models were verified through cross-validation. The variable representing the elevation of the ore unit base was then estimated at the unknown locations using the prediction data-set. The estimated values were compared against the validation data using mean absolute error (MAE) and mean squared error (MSE) criteria. The results show although IRF-k slightly outperformed OK and UK, all the algorithms produced satisfactory and similar results. MSE values obtained from the comparison with the validation data were 0.1267, 0.1322, and 0.1349 for IRF-k, OK, and UK algorithms respectively. The similarity in the results generated by these algorithms is explained by the existence of a large data-set and the chosen neighbourhood parameters for the kriging technique

Universality, Scaling and Triviality in a Hierarchical Field Theory

Using polynomial truncations of the Fourier transform of the RG transformation of Dyson's hierarchical model, we show that it is possible to calculate very accurately the renormalized quantities in the symmetric phase. Numerical results regarding the corrections to the scaling laws, (i.e finite cut-off dependence) triviality, hyperscaling, universality and high-accuracy determinations of the critical exponents are discussed.Comment: LATTICE98(spin

Non-Gaussian numerical errors versus mass hierarchy

We probe the numerical errors made in renormalization group calculations by varying slightly the rescaling factor of the fields and rescaling back in order to get the same (if there were no round-off errors) zero momentum 2-point function (magnetic susceptibility). The actual calculations were performed with Dyson's hierarchical model and a simplified version of it. We compare the distributions of numerical values obtained from a large sample of rescaling factors with the (Gaussian by design) distribution of a random number generator and find significant departures from the Gaussian behavior. In addition, the average value differ (robustly) from the exact answer by a quantity which is of the same order as the standard deviation. We provide a simple model in which the errors made at shorter distance have a larger weight than those made at larger distance. This model explains in part the non-Gaussian features and why the central-limit theorem does not apply.Comment: 26 pages, 7 figures, uses Revte