The 1/N expansion of colored tensor models

In this paper we perform the 1/N expansion of the colored three dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with more and more complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S^3 contribute to the leading order in the large N limit.Comment: typos corrected, references update

Group field theories

Group field theories are particular quantum field theories defined on D copies of a group which reproduce spin foam amplitudes on a space-time of dimension D. In these lecture notes, we present the general construction of group field theories, merging ideas from tensor models and loop quantum gravity. This lecture is organized as follows. In the first section, we present basic aspects of quantum field theory and matrix models. The second section is devoted to general aspects of tensor models and group field theory and in the last section we examine properties of the group field formulation of BF theory and the EPRL model. We conclude with a few possible research topics, like the construction of a continuum limit based on the double scaling limit or the relation to loop quantum gravity through Schwinger-Dyson equationsComment: Lectures given at the "3rd Quantum Gravity and Quantum Geometry School", march 2011, Zakopan

Renormalization of an Abelian Tensor Group Field Theory: Solution at Leading Order

We study a just renormalizable tensorial group field theory of rank six with quartic melonic interactions and Abelian group U(1). We introduce the formalism of the intermediate field, which allows a precise characterization of the leading order Feynman graphs. We define the renormalization of the model, compute its (perturbative) renormalization group flow and write its expansion in terms of effective couplings. We then establish closed equations for the two point and four point functions at leading (melonic) order. Using the effective expansion and its uniform exponential bounds we prove that these equations admit a unique solution at small renormalized coupling.Comment: 37 pages, 14 figure

Multiscale statistical process control with multiresolution data

An approach is presented for conducting multiscale statistical process control that adequately integrates data at different resolutions (multiresolution data), called MR-MSSPC. Its general structure is based on Bakshi's MSSPC framework designed to handle data at a single resolution. Significant modifications were introduced in order to process multiresolution information. The main MR-MSSPC features are presented and illustrated through three examples. Issues related to real world implementations and with the interpretation of the multiscale covariance structure are addressed in a fourth example, where a CSTR system under feedback control is simulated. Our approach proved to be able to provide a clearer definition of the regions where significant events occur and a more sensitive response when the process is brought back to normal operation, when it is compared with previous approaches based on single resolution data. © 2006 American Institute of Chemical Engineers AIChE J, 200