Renormalization of the 2-point function of the Hubbard model at half-filling

We prove that the two dimensional Hubbard model at finite temperature T and half-filling is analytic in the coupling constant in a radius at least $c/(\log T)^2$. We also study the self-energy through a new two-particle irreducible expansion and prove that this model is not a Fermi liquid, but a Luttinger liquid with logarithmic corrections. The techniques used are borrowed from constructive field theory so the result is mathematically rigorous and completely non-perturbative. Together with earlier results on the existence of two dimensional Fermi liquids, this new result proves that the nature of interacting Fermi systems in two dimensions depends on the shape of the Fermi surface.Comment: 45 pages, 28 figure

The Hubbard model at half-filling, part III: the lower bound on the self-energy

We complete the proof that the two-dimensional Hubbard model at half-filling is not a Fermi liquid in the mathematically precise sense of Salmhofer, by establishing a lower bound on a second derivative in momentum of the first non-trivial self-energy graph.Comment: 31 pages, 4 figure

Ward type identities for the 2d Anderson model at weak disorder

Using the particular momentum conservation laws in dimension d=2, we can rewrite the Anderson model in terms of low momentum long range fields, at the price of introducing electron loops. The corresponding loops satisfy a Ward type identity, hence are much smaller than expected. This fact should be useful for a study of the weak-coupling model in the middle of the spectrum of the free Hamiltonian.Comment: LaTeX 2e document using AMS symbols, 25 pages and 32 eps figure

Scaling behaviour of three-dimensional group field theory

Group field theory is a generalization of matrix models, with triangulated pseudomanifolds as Feynman diagrams and state sum invariants as Feynman amplitudes. In this paper, we consider Boulatov's three-dimensional model and its Freidel-Louapre positive regularization (hereafter the BFL model) with a ?ultraviolet' cutoff, and study rigorously their scaling behavior in the large cutoff limit. We prove an optimal bound on large order Feynman amplitudes, which shows that the BFL model is perturbatively more divergent than the former. We then upgrade this result to the constructive level, using, in a self-contained way, the modern tools of constructive field theory: we construct the Borel sum of the BFL perturbative series via a convergent ?cactus' expansion, and establish the ?ultraviolet' scaling of its Borel radius. Our method shows how the ?sum over trian- gulations' in quantum gravity can be tamed rigorously, and paves the way for the renormalization program in group field theory

From constructive field theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics

Let $B=(B_1(t),..,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\alpha\le 1/4$, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of $B$ is a difficult task because of the low H\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We intend to show in a forthcoming series of papers how to desingularize iterated integrals by a weak singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using "standard" tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates of the moments and call for an extension of the Gaussian tools such as for instance the Malliavin calculus. This first paper aims to be both a presentation of the basics of rough path theory to physicists, and of perturbative field theory to probabilists; it is only heuristic, in particular because the desingularization of iterated integrals is really a {\em non-perturbative} effect. It is also meant to be a general motivating introduction to the subject, with some insights into quantum field theory and stochastic calculus. The interested reader should read in a second time the companion article \cite{MagUnt2} (or a preliminary version arXiv:1006.1255) for the constructive proofs

SYSTEMES HYBRIDES ET ANNOTATION RECIPROQUE MISE A DISPOSITION... MISE EN DISPOSITION

En tentant de caractériser en quoi deux ingénieries en cours au sein du Centre de Recherche sur l'Education, les Apprentissages et la Didactique (EA3875), les auteurs espèrent apporter un nouveau regard sur la façon dont l'instrumentation peut et doit orienter la manière de concevoir les dispositifs ingénieriques. La notion de système hybride sera particulièrement étudiée et illustrée dans le but de nourrir une réflexion sur les nouveaux régimes du voir et du comprendre en Sciences de l'Educatio

Bosonic Colored Group Field Theory

Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the "ultraspin" (large spin) limit. The results are generalized in any dimension. Finally integrating out two colors we write a new representation which could be useful for the constructive analysis of this type of models

Vanishing of Beta Function of Non Commutative Φ44\Phi^4_4 Theory to all orders

The simplest non commutative renormalizable field theory, the $\phi_4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe up to three loops, as shown by H. Grosse and R. Wulkenhaar, M. Disertori and V. Rivasseau. We extend this result to all orders.Comment: 12 pages, 3 figure