Summer School in Mathematical Physics

Rigorous Results in Statistical Mechanics
and
Quantum Field Theory

Lectures

Anton Bovier

Introduction to the Statistical Mechanics of Spin Glasses

Contents:

1.Gibbs measures for disordered spin systems; the metastate concept.
2.Example: The random energy model
3.Spin glasses and gaussian processes; geometry of random measures
4.Comparison methods and Parisi solution

Lecture notes

Abstract: This lecture series will give a brief introduction to the main concepts from statistical mechanics with a special emphasis on disordered systems in general, and of spin glasses in particular. Besides the mathematical tools, the physical origin of the models will be discussed. A full treatment of simple models such as the REM will be given, and the complex structures emerging in the GREM will be indicated. Finally, the interpolation and comparison methods that have led to a rigorous proof of the Parisi solution in the Sherrington-Kirkpatrick model will be explained.

François Dunlop

An Introduction to Rigorous Statistical Mechanics

Contents:
1.Gibbs formalism
2.Cluster expansions
3.Bulk and surface transitions
4.Non-gibbsianness

Lecture notes

Bertrand Duplantier

Random Conformal Geometry and Quantum Gravity

Abstract: Conformally invariant random paths are ubiquitous in many two-dimensional statistical models, like the Ising and Potts models, and in fundamental stochastic processes such as Brownian motion. Their universal geometrical properties will be described in relation to two-dimensional quantum gravity.

Lecture 1 / Lecture 2 / Lecture 3 / Lecture 4

Jacques Magnen

Introduction to constructive field theory.

- functional integrals
- correlation functions
- perturbation series, its divergence.
- introduction of momentum and space cutoffs.
- Cluster expansion in a single scale: standard form and new approach.
- introduction to the multiscale analysis and renormalisation i.e. to the renormalisation group.

References:
On the construction of the Gross-Neveu model:
arXiv:hep-th/9802145 Title: Continuous Constructive Fermionic Renormalization Authors: M. Disertori, V. Rivasseau
On the construction of the fi4 model in one scale:
arXiv:0706.2457v1 [math-ph] Title: Constructive $\phi^4$ field theory without tears Authors: J. Magnen, V. Rivasseau

Vincent Rivasseau

Multiscale Analysis and Renormalization of the $\phi^4_4$ Quantum Field Theory

Abstract: We shall explain first why renormalization is best analyzed through multiscale analysis, taking as example the $\phi^4_4$ Euclidean field theory. We shall then proceed to explain how to compute the renormalization group flow of the coupling constant. Unfortunately the famous "triviality" or "Landau ghost" does not allow to construct the ultraviolet limit of that model in a nonperturbative sense. We shall then introduce and study the noncommutative Grosse-Wulkenhaar model, a $\phi^{\star 4}_4$ Euclidean field theory on the Moyal space which is free of this Landau ghost or triviality problem, hence can be fully built in the ultraviolet limit.

Lecture 1 / Lecture 2 / Lecture 3 / Lecture 4

Roland Sénéor

Constructive Field Theories and Probability Methods

Abstract: It is possible to express field theories in Euclidean space as the invariant measures of some stochastic differential system. A stochastic analysis approach to field theories will be progressively developed showing the similarities and the differences between field theory concepts and probabilistic ideas. Such an analysis will naturally lead to applications to the study of classical dissipative systems.

Gordon Slade

Introduction to Percolation Theory

Abstract: Percolation theory is a basic example of critical phenomena in statistical mechanics, with qualitative and quantitative connections to other models such as the Ising model of ferromagnetism. Its very rich mathematical structure has led it to become a central object of study in modern probability theory. This course will provide an introduction to the mathematics of percolation theory, with emphasis on its critical behaviour.

References: S. Smirnov, Critical Percolation in the Plane
G. Grimmett, Percolation, 2nd edition, Springer (1999)