EDIPS

The dynamics of extremes/outliers, i.e., particles that live at the edge of the crowd, often controls the time evolution of many-body complex systems. These include bacterial infections, barrier crossing in chemical reactions, search processes, signal transmission in biology and many others. While the classical extreme value statistics provides a framework to understand the distribution of extremes for non-interacting particles at a fixed time, there is no theory on how this extreme evolves with time or on the universality of the multi-time correlation functions. In fact, in the presence of interactions and/or disorder, there are no general quantitative methods to study the dynamics of extremes. The ambitious goal of EDIPS is to develop a theory to describe the temporal evolution of extremes in models of strongly interacting particles (short and long-ranged, with or without disorder), with applications to more complex systems in non-linear dynamics, soft matter, probability, climatology, and finance. This will be achieved by developing analytical tools such as large deviation theory, hydrodynamics, integrability and random matrix theory, as well as numerical methods involving importance sampling. We will start by studying simple, yet nontrivial, models where exact results can be obtained. We will then use these insights to develop a mathematical toolbox to describe the dynamics of extremes in more complex systems. EDIPS is divided into four main tasks. In Task 1 we will investigate the extreme dynamics for diffusive Langevin processes with and without interactions. We will compute two-time, or more generally multi-time correlation functions of the extreme, in one or higher dimensions. In Task 2 we will consider a system of N diffusive particles subject to stochastic resetting to their initial positions at random times. This simultaneous resetting makes them strongly correlated and drives the system into a non-equilibrium stationary state. Here also, we will compute the multi-time correlation functions of the extreme in all dimensions. In Task 3 we will address the diffusion of outliers in time dependent random media, with the aim to characterize how the common environment affects the statistics of the outliers in one and higher dimensions. This is important for current experiments using tracer diffusion in fluids designed to test the violations of Einstein's diffusion picture. In Task 4 we will apply the methods developed in Tasks 1-3 to the problem of diffusion of charged interacting ions in channels to understand how the dynamics of outliers control transport properties through the channels. Understanding the effect of the surface charge distributions on the current flowing in the channel will be an important advance in the field of ionic transport under confinement. These four tasks illustrate the breadth of the spectrum of problems where the dynamics of extremes play a major role, with an abundance of applications. EDIPS will demonstrate the complementarity of these four tasks and provide a unifying theory for the dynamics of extremes. It will provide a new theoretical framework for a variety of problems, new algorithms to measure large deviations in other contexts, and stimulate new experiments. The consortium is composed of four theory teams (LPTMS, LPENS, LPTHE, LOMA) who have extensively collaborated in the past on statistical physics, condensed matter and soft matter physics. They have contributed to recent progresses on extremes in different contexts. Given their track record of a broad spectrum of international collaborations and their complementary expertise in these topics, they are in a unique position to make this ambitious project a success, with important impact in a variety of fields, which will be highlighted by the organization of a workshop. We request the recruitment of a postdoc for two years, one year at LPTMS and one year at LPENS to increase synergy between the teams.