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

Polchinski's exact renormalisation group for tensorial theories: Gaussian universality and power counting

In this paper, we use the exact renormalisation in the context of tensor models and tensorial group field theories. As a byproduct, we rederive Gaussian universality for random tensors and provide a general power counting for Abelian tensorial field theories with a closure constraint, leading us to a only five renormalizable theories.Comment: 22 pages, 4 figure

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

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.

Wilsonian renormalization, differential equations and Hopf algebras

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $\phi^{3}$ diagrams and a certain class of decorated rooted trees.Comment: 42 pages, 26 figures in PDF format, extended version of a talk given at the conference "Combinatorics and physics" held at Max Planck Institut fuer Mathematik in Bonn in march 2007, some misprints correcte

Exact Renormalisation Group Equations and Loop Equations for Tensor Models

In this paper, we review some general formulations of exact renormalisation group equations and loop equations for tensor models and tensorial group field theories. We illustrate the use of these equations in the derivation of the leading order expectation values of observables in tensor models. Furthermore, we use the exact renormalisation group equations to establish a suitable scaling dimension for interactions in Abelian tensorial group field theories with a closure constraint. We also present analogues of the loop equations for tensor models

Constructive Matrix Theory for Higher Order Interaction

This paper provides an extension of the constructive loop vertex expansion to stable matrix models with interactions of arbitrarily high order. We introduce a new representation for such models, then perform a forest expansion on this representation. It allows to prove that the perturbation series of the free energy for such models is analytic in a domain uniform in the size N of the matrix. Our method applies to complex (rectangular) matrices. The extension to Hermitian square matrices, which was claimed wrongly in the first arXiv version of this paper, is postponed to a future study.Comment: 44 pages, 9 figure