V. Rivasseau

Trees, forests and jungles: a botanical garden for cluster expansions

Combinatoric formulas for cluster expansions have been improved many times over the years. Here we develop some new combinatoric proofs and extensions of the tree formulas of Brydges and Kennedy, and test them on a series of pedagogical examples.

A renormalizable field theory: the massive Gross-Neveu model in two dimensions

The Euclidean massive Gross-Neveu model in two dimensions is just renormalizable and asymptotically free. Thanks to the Pauli principle, bare perturbation theory with an ultra-violet cut-off (and the correct ansatz for the bare mass) is convergent in a disk, whose radius corresponds by asymptotic freedom to a small finite renormalized coupling constant. Therefore, the theory can be fully constructed in a perturbative way. It satisfies the O.S. axioms and is the Borel sum of the renormalized perturbation expansion of the model

Construction and Borel summability of infrared Φ 4 4 by a phase space expansion

We construct the thermodynamic limit of the critical (massless) φ4 model in 4 dimensions with an ultraviolet cutoff by means of a “partly renormalized” phase space expansion. This expansion requires in a natural way the introduction of effective or “running” constants, and the infrared asymptotic freedom of the model, i.e. the decay of the running coupling constant, plays a crucial rôle. We prove also that the correlation functions of the model are the Borel sums of their perturbation expansion.

Local existence of the Borel transform in Euclidean φ 4 4

We bound rigorously the large order behaviour of φ44 euclidean perturbative quantum field theory, as the simplest example of renormalizable, but non-super-renormalizable theory. The needed methods are developed to take into account the structure of renormalization, which plays a crucial role in the estimates. As a main thorem, it is shown that the Schwinger functions at ordern are bounded byKnn!, which implies a finite radius of convergence for the Borel transform of the perturbation series.

On the microscopic validity of the Wulff construction and of the generalized Young equation

For a large class of 1+1 dimensional interfaces of the Solid-On-Solid type we prove on a microscopic basis the validity of the Wulff construction and of the generalized Young equation which gives the contact angle of a sessile drop on a wall. Our proof relies on a new method to treat random walks with a finite number of global constraints.

Construction and Borel summability of planar

We use the methods of [1] to show that the planar part of the renormalized perturbation theory forϕ44-euclidean field theory is Borel-summable on the asymptotically free side of the theory. The Borel sum can therefore be taken as a rigorous definition of theN→∞ limit of a massiveN×N matrix model with a +trgϕ4 interaction, hence with “wrong sign” ofg. Our construction is relevant for a solution of the ultra-violet problem for planar QCD. We also propose a program for studying the structure of the “renormalons” singularities within the planar world.

4

We provide the basis for a rigorous construction of the Schwinger functions of the pure SU(2) Yang-Mills field theory in four dimensions (in the trivial topological sector) with a fixed infrared cutoff but no ultraviolet cutoff, in a regularized axial gauge. The construction exploits the positivity of the axial gauge at large field. For small fields, a different gauge, more suited to perturbative computations is used; this gauge and the corresponding propagator depends on large background fields of lower momenta. The crucial point is to control (in a non-perturbative way) the combined effect of the functional integrals over small field regions associated to a large background field and of the counterterms which restore the gauge invariance broken by the cutoff. We prove that this combined effect is stabilizing if we use cutoffs of a certain type in momentum space. We check the validity of the construction by showing that Slavnov identities (which express infinitesimal gauge invariance) do hold non-perturbatively.

-dimensional Euclidean field theory

We prove that in a two-dimensional Gaussian SOS model with a small attractive potential the height of the interface remains bounded no matter how small the potential is; this is in sharp contrast with the free situation in which the interface height diverges logarithmically in the thermodynamic limit.

Construction of

We introduce a new type of cluster expansion which generalizes a previous formula of Brydges and Kennedy. The method is especially suited for performing a phase-space multiscale expansion in a just renormalizable theory, and allows the writing of explicit non-perturbative formulas for the Schwinger functions. The procedure is quite model independent, but for simplicity we chose the infrared

{\rm YM}_4

\ph^4_4