1. BackgroundThe Faculty of Law of the Universidad de Buenos Aires, ranked as one the most important in Latin America, focuses its academic activities on “Human Rights” and “Integration Law”. In particular, the Faculty of Law is the headquarters of Jean Monnet Chair, held by Professor C.Pizzolo, which is very active in organizing dozens of academic activities in the countries of Mercosur and within European Union. 2. Rationale and ContextWithin this context, both the University of Buenos Aires and its Faculty of Law are fully committed to the development of the teaching and research activities concerning the role of the human rights in the integration process in Latin America, taking into account the experience of EU. As matter a fact the EU, initially born as economic integration process, early turn into a legal system protecting individual rights (see the judgments Van Gen den Los of 1963 and Stauder of the 1969; and today the Article 6 of the Treaty on European Union and the Charter of the Fundamental Rights of EU). The possibility of the individuals to claim the protection their rights, arising form the integration process, has made them the main “Guardian” in the actual implementation of the EU law as argued by J.H.H. Weiler.The establishment of a prestigious Jean Monnet Centre of Excellence would be the instrument in order to strengthen those activities and to constitute an important reference in Latin America.

Phase transitions are ubiquitous in systems consisting of a large number of interacting components, which can be as simple as a gas or as complex as human society. Often one observes intriguing associated dynamic phenomena such as a separation of time scales and metastability. A classical example is the liquid-gas transition of water, which exhibits metastability of supersaturated vapour over relatively long time scales, followed by a rapid transition to the liquid phase. A phenomenological description of such metastable states goes back to van der Waals theory of non-ideal gases. Besides a proper description of the states, the relevant dynamic aspects are the lifetimes of the states and how transitions occur between them. In reality transitions are often triggered by small impurities (causing e.g. droplet nucleation in vapour), and in mathematical models this is often achieved by adding randomness to dynamics. Stochastic particle systems, where idealized particles move and interact in a discrete (lattice) geometry, provide therefore a very natural class of models to study and understand the dynamics of such transitions and the concept of metastability.A mathematically rigorous approach poses very challenging research questions, and is an active area of modern probability theory where significant recent progress has been achieved. The proposed research builds on these developments and aims towards a full rigorous understanding of a condensation transition in zero-range processes, a particular class of stochastic particle systems which has attracted recent research interest in theoretical physics. Condensation here means, that with increasing density the system switches from a homogeneous distribution of particles to a state where a macroscopic fraction of all the particles condenses on a single lattice site. Zero-range processes are therefore used as generic models of condensation phenomena with applications ranging from clustering of granular materials to the formation of giant hubs in complex network dynamics. They are also of theoretical interest as effective models of domain wall dynamics separating different phases in more general systems, explaining phenomena like the formation of traffic jams on highways.Zero-range processes show a very rich critical behavior with interesting dynamic phenomena on several time scales including metastability, which have been understood on a heuristic level in statistical physics, inspiring many ideas in this proposal. The aim of the project is to underpin these findings with rigorous probabilistic results by proving scaling limits for the dynamics of effective observables on several different time scales. These concrete outcomes will be put into a wider context and will lead to methodological advances, by improving and generalizing recent mathematical techniques and understanding the exact conditions of validity of heuristic arguments used for predictions. Also conceptual insights are invisaged, by exploring new approaches for the mathematical characterization of metastability phenomena in stochastic particle systems.

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