A renormalisation group approach to the universality of Wigner's semicircle law for random matrices with dependent entries

We show that if the non Gaussian part of the cumulants of a random matrix model obey some scaling bounds in the size of the matrix, then Wigner's semicircle law holds. This result is derived using the replica technique and an analogue of the renormalisation group equation for the replica effective action. This is a transcript of a talk given at "5th Winter Workshop on Non-Perturbative Quantum Field Theory" Sophia-Antipolis, March 2017 and is based on former work in collaboration with A. Tanasa and D.L. Vu, see arXiv:1609.01873 .Comment: 12 pages, 4 figures, to appear in the conference proceeding

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

Schwinger-Dyson Equations in Group Field Theories of Quantum Gravity

In this talk, we elaborate on the operation of graph contraction introduced by Gurau in his study of the Schwinger-Dyson equations. After a brief review of colored tensor models, we identify the Lie algebra appearing in the Schwinger-Dyson equations as a Lie algebra associated to a Hopf algebra of the Connes-Kreimer type. Then, we show how this operation also leads to an analogue of the Wilsonian flow for the effective action. Finally, we sketch how this formalism may be adapted to group field theories.Comment: talk given at "The XXIX International Colloquium on Group-Theoretical Methods in Physics", Chern Institute of Mathematics August 2012, submitted to the conference proceeding

Legitimizing global economic governance through transnational parliamentarization: The parliamentary dimensions of the WTO and the World Bank

This paper discusses the potential contribution of parliamentary institutions and networks to the democratization of global economic governance. It places the analysis in the context of the larger debate on the democratic deficit of international economic institutions, in particular the WTO. On a theoretical level, the paper distinguishes different notions of legitimacy and democracy in order to identify which aspects of democratic legitimacy of global economic governance can be addressed through transnational parliamentarization. It is argued that national parliaments must react to the emergence of global economic governance in a multi-level system through new forms of transnational parliamentarization. In its empirical part, the paper assesses the Parliamentary Conference on the WTO (PCWTO) and the Parliamentary Network on the World Bank (PNoWB) as two examples of such transnational parliamentarization. Drawing on the theory of deliberative democracy the paper argues that the contribution of these settings to democratic global governance should not be measured on the basis of their formal decision-making power but with regard to their role as fora for transnational discourses and on their potential to empower national parliamentarians. -- Das Arbeitspapier untersucht den potentiellen Beitrag von parlamentarischen Institutionen und Netzwerken zur Demokratisierung des internationalen Wirtschaftssystems. Es stellt die Analyse in den Kontext der allgemeinen Debatte über das Demokratiedefizit der globalen Wirtschaftsinstitutionen, insbesondere der WTO. Auf theoretischer Ebene unterscheidet das Arbeitspapier verschiedene Konzeptionen von Legitimität und Demokratie, um herauszufiltern, welche Anforderungen demokratischer Legitimität durch Prozesse der transnationalen Parlamentarisierung erfüllt werden können. Das Arbeitspapier argumentiert, dass nationale Parlamente auf die Emergenz globaler Wirtschaftsinstitutionen im Mehrebenensystem durch neue Formen der transnationalen Parlamentarisierung reagieren müssen. In seinem empirischen Teil bewertet das Arbeitspapier die Parlamentarische Konferenz zur WTO (Parliamentary Conference on the WTO, PCWTO) und das Parlamentarische Netzwerk zur Weltbank (Parliamentary Network on the World Bank, PNoWB) als zwei Beispiele einer derartigen transnationalen Parlamentarisierung. Unter Rückgriff auf die Theorie der deliberativen Demokratie wird argumentiert, dass der Beitrag dieser Einrichtungen zu demokratischer gobal governance nicht anhand ihrer formalen Entscheidungsbefugnisse gemessen werden sollte, sondern mit Blick auf ihre Rolle als Foren für transnationale Diskurse und ihr Potential nationale Parlamentarier zu stärken.

On Kreimer's Hopf algebra structure of Feynman graphs

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements.Comment: 21 pages, LaTeX2e, requires feynmf package to draw Feynman graphs (see log file for additional information). Following an idea by Dirk Kreimer we introduced in the revised version a primitivator which maps overlapping divergences to primitive elements and which provides the link to the Hopf algebra of Kreimer (q-alg/9707029, hep-th/9808042). v4: error in eq (29) corrected and references updated; to appear in Eur.Phys.J.

Analyticity results for the cumulants in a random matrix model

The generating function of the cumulants in random matrix models, as well as the cumulants themselves, can be expanded as asymptotic (divergent) series indexed by maps. While at fixed genus the sums over maps converge, the sums over genera do not. In this paper we obtain alternative expansions both for the generating function and for the cumulants that cure this problem. We provide explicit and convergent expansions for the cumulants, for the remainders of their perturbative expansion (in the size of the maps) and for the remainders of their topological expansion (in the genus of the maps). We show that any cumulant is an analytic function inside a cardioid domain in the complex plane and we prove that any cumulant is Borel summable at the origin

Logic and the Concept of God

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions to the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago

Stochastic linear scaling for metals and non metals

Total energy electronic structure calculations, based on density functional theory or on the more empirical tight binding approach, are generally believed to scale as the cube of the number of electrons. By using the localisaton property of the high temperature density matrix we present exact deterministic algorithms that scale linearly in one dimension and quadratically in all others. We also introduce a stochastic algorithm that scales linearly with system size. These results hold for metallic and non metallic systems and are substantiated by numerical calculations on model systems.Comment: 9 pages, 2 figure