Statistical mechanics: contributions to rigidity percolation and long range interacting systems

This dissertation presents my scientific activities since the end of my PhD in 2003. During this period, I spent two years as a post doc at the Los Alamos National Laboratory (USA), in the Center for Non Linear Studies (CNLS) and the Condensed Matter group. Then, in Fall 2005, I was recruited as a ”Maˆıtre de Conf ́erences” at the Nice-Sophia Antipolis University, in the Mathematics Department.At first glance, my activity can be divided in two rather distinct themes: rigidity percolation on one side, and long range interacting systems on the other side, the latter regrouping a rather diverse body of works. However, my works in these two directions have several features in common. First, at the level of methods. In both cases, the problem is usually to understand some macroscopic properties starting from a microscopic modeling, using probabilistic tools: this might be a definition of the statistical mechanics endeavor in general. More precisely, tools from large deviation theory play an important role in many of the works presented here, both for rigidity percolation and long range interacting systems. Second, there are similarities in the strategies to attack the problems: I have often concentrated first on simple models, on which a detailed study is possible, before trying to extract a generic behavior.Although the statistical mechanics of long range interacting systems was already the subject of my PhD thesis, done under the joint supervision of Thierry Dauxois and Stefano Ruffo, my research on the subject has followed new directions since 2003: my position in a mathematics department gave me the opportunity to start a mathematically oriented research project on ki- netic limits for systems of interacting particles, with Pierre-Emmanuel Jabin and Maxime Hauray; at the same time, I started a collaboration with an experimental team on cold atoms in INLN (Institut Non Lin ́eaire de Nice).The dissertation is organized in two chapters: the first one is devoted to rigidity percolation, and the second one to long range interacting systems. In each case, I have tried to give a detailed and non technical introduction to the subject, to emphasize the motivations and questions behind my works, andto present and summarize my contributions. I then append to each chapter a few articles which I consider to best illustrate my scientific activity since 2003.There is one bibliography at the end of each chapter. Citations of the type [B*] refer to works of which I am a coauthor. These references are gathered at the end of the document for clarity.The works presented in this dissertation owe much to my numerous col- laborators during these years; I take this opportunity to warmly thank all of them