AMP algorithms and Stein's method: Understanding TAP equations with a new method

We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available.Comment: 38 page

On concavity of TAP free energy in the SK model

We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line, evaluated in Bolthausen's approximate solutions of the TAP equations. We show that while its empirical spectrum weakly converges to a measure with negative support, positive outlier eigenvalues occur for some $(\beta,h)$ below the AT line. In this sense, TAP free energy may lose concavity in the order parameter of the theory, i.e. the random spin-magnetisations, even below the AT line. Possible interpretations of these findings within Plefka's expansion of the Gibbs potential are not definitive and include the following: i) either higher order terms shall not be neglected even if Plefka's first convergence criterion (yielding, in infinite volume, the AT line) is satisfied, ii) Plefka's first convergence criterion (hence the AT line) is necessary yet hardly sufficient, or iii) Bolthausen's magnetizations do not approximate the TAP solutions sufficiently well up to the AT line.Comment: 29 pages, 1 figur

French Roadmap for complex Systems 2008-2009

This second issue of the French Complex Systems Roadmap is the outcome of the Entretiens de Cargese 2008, an interdisciplinary brainstorming session organized over one week in 2008, jointly by RNSC, ISC-PIF and IXXI. It capitalizes on the first roadmap and gathers contributions of more than 70 scientists from major French institutions. The aim of this roadmap is to foster the coordination of the complex systems community on focused topics and questions, as well as to present contributions and challenges in the complex systems sciences and complexity science to the public, political and industrial spheres

First passage percolation in the mean field limit

This dissertation deals with two classical problems in statistical mechanics: the first passage percolation on Euclidean spaces, FPP for short, in both directed and undirected settings

Statistical Mechanics of Classical and Disordered Systems

International audienc

French Roadmap for complex Systems 2008-2009

This second issue of the French Complex Systems Roadmap is the outcome of the Entretiens de Cargese 2008, an interdisciplinary brainstorming session organized over one week in 2008, jointly by RNSC, ISC-PIF and IXXI. It capitalizes on the first roadmap and gathers contributions of more than 70 scientists from major French institutions. The aim of this roadmap is to foster the coordination of the complex systems community on focused topics and questions, as well as to present contributions and challenges in the complex systems sciences and complexity science to the public, political and industrial spheres