Mini cours par Erwin Bolthausen : The Thouless-Anderson-Palmer equation in spin glass theory.
Notes de cours détaillées (version finale) Notes.pdf
Thouless, Anderson and Palmer proposed very early in 1977 an approach for the analysis of spin glasses, in particular the Sherrington-Kirkpatrick model, which predated the Parisi approach, and which was later shadowed by the latter. However, the TAP equations played a considerable role in the physics literature on the topic, and it has recently become evident that they are of interest also in the mathematical analysis of spin glass models.
The minicourse does not require any previous knowledge of spin glasses.
A summary of the topics:
Lect 1: The belief propagation equations. These equations are widely applied in a number of different fields, like in coding theory, or in theoretical computer science. The equations give a fast way to compute marginal distributions in the case where the models are defined on a tree graph. Somewhat vaguely stated, "replica symmetry" is equivalent with the property that in large systems, the equations approximatively compute the marginal distributions.
Lect 2: Heuristic derivation of the TAP equations from the belief propagation equations: This is perhaps the easiest way to derive (but not rigorously prove) the TAP equations in a number of models, like in the SK-model, the perceptron, or the Hopfield net. Probably, we also discuss a number of numerical applications of the TAP equations. In fact, if available, they are numerically easier to handle than the belief propagation equations.
Lect 3: The direct iterative construction of solutions of the TAP equations: Originally, it was believed that a direct mathematical construction of solutions of the equations is quite difficult, even in the replica symmetric regime. This has changed recently, and we present an approach which constructs such solutions in the full high temperature regime of the SK-model, and in the perceptron model.
Lect 4: As a mathematical application of the TAP equations, and their iterative construction, we give a new proof of Gardner's formula for the perceptron. This part is still a bit tentative, and perhaps we restrict that to the simpler case of the SK-model. Essentially, the method is a sophisticated version of the classical second moment method.
Mini cours par Christophe Sabot : Linearly reinforced processes and related topics
We will present a panorama of recent developments about the Edge Reinforced Random Walk, the Vertex Reinforced Jump Process and related objects as supersymmetric sigma-fields and random Schrödinger operators. In particular, we will present the following aspects:
- exangeability: we will explain the ideas behind the representation of these processes as mixtures of Markov processes and how it is related to explicite computations of some integrals.These explicite computations can be understood either by probabilistic considerations or internal symmetries of the integrand.
- localization/delocalization estimates: we will give an idea about the arguments behind the 3 estimates that play a crucial role in the subject (exponential localization at strong disorder, delocalization at weak disorder in d>2, and polynomial localization in d=2).
- We will explain how the asymptotic behavior of the process is related to the properties of a random Schrödinger operators.
Exposés :
Elie Aidekon : Points of infinite multiplicity of a planar Brownian motion.
Points of infinite multiplicity are particular points which the Brownian motion visits infinitely often.Following a work of Bass, Burdzy and Khoshnevisan, we construct and study a measure carried by these points.
Joint work with Yueyun Hu and Zhan Shi.
Pierre Andreoletti : Favorite range for recurrent biased random walks on trees
We study the asymptotic of the log-number of edges sufficiently visited by recurrent biased random walks on trees.
It turns out that slow and sub-diffusive cases present a common behavior whereas the diffusive case depends on its specific parameter. We also try to make a link between this behavior and the fluctuations of the underlying environment.
Amine Asselah : Deviations Estimates for the Capacity of a Random Walk in d>4.
We focus on the ways a simple random walkreduces the capacity of its range, when dimension is larger than 4.We look at large and moderate deviations.We discuss a robust method, its strengh and weakness.
This is a joint work with Bruno Schapira.
Oriane Blondel : Marches aléatoires en environnement dynamique à décorrélation polynomiale en dimension 1
On étudie des marches aléatoires dans des environnements stationnaires dynamiques en dimension 1.
Sous une hypothèse de décroissance polynomiale (avec exposant assez grand) de la covariance entre
des boîtes bien séparées en temps, on montre une loi des grands nombres pour une classe assez générale de marches aléatoires aux plus proches voisins.
Travail en commun avec Marcelo Hilario et Augusto Teixeira.
Jean-Baptiste Gouéré : Percolation et percolation de premier passage dans le modèle booléen.
Yueyun Hu : The effective resistances of some random trees.
Tobias Hurth : Exponential ergodicity for piecewise deterministic Markov processes
In this talk, we consider a certain class of piecewise deterministic Markov processes characterized by Poissonian switching between deterministic vector fields. We discuss sufficient conditions for exponential convergence in total variation to the invariant measure of the associated Markov semigroup, and illustrate these conditions through several examples. The talk is based on work with Michel Benaïm and Edouard Strickler.
Michel Pain : Fluctuations of the critical Gibbs measure of branching Brownian motion
The Gibbs measure of branching Brownian motion has atoms at position of the particles, with a Gibbs weight according to this position. At the critical temperature, this measure captures the form of the front of branching Brownian motion, and it was proved by Madaule that it converges after rescaling to a marginal of a Brownian meander times the limit of the derivative martingale.
I will present some recent results obtained with Pascal Maillard, where we obtain the fluctuations in this convergence, with 1-stable distributions as possible limits. This applies in particular to the additive and derivative martingales, with respectively a totally asymmetric and a Cauchy distributions in the limit.
Françoise Pène : Random walks in random sceneries and related models.
Rémy Poudevigne : Limit theorem for sub-ballistic random walk in random environment.
Pierre Rousselin : Chute de dimension pour les marches aléatoires sur les arbres aléatoires.
Nous nous intéressons à différents modèles d'arbres aléatoires et aux marches aléatoires sur les sommets de tels arbres. Dans le cas où la marche aléatoire est transiente, la marche partpresque sûrement vers l'infini en empruntant un rayon aléatoire. La loi de ce rayon est appelée la mesure harmonique sur le bord de l'arbre. Un phénomène de chute de dimension se produit : cette mesure harmonique est presque sûrement concentrée sur une partie petite (au sens de ladimension de Hausdorff) du bord de l'arbre. Autrement dit, avec grande probabilité, les trajectoires de la marche aléatoires sont presque sûrement comprises dans un sous-arbre beaucoup plus fin que l'arbre original. Cette théorie a été initiée par Russel Lyons, Robin Pemantle et Yuval Peres dans les années 1990. Plus récemment, Nicolas Curien, Jean-François Le Gall, puis Shen Lin ont étudié ce phénomène sur un autre modèle d'arbres aléatoires. Nous rappellerons leurs résultats et discuteront des généralisations (https://arxiv.org/abs/1708.06965 et https://arxiv.org/abs/1711.07920) sur lesquelles nous avons travaillé.
Olivier Zindy : Universality of the critical density for activated random walk.
