Interacting Random Walks – MALIN

Interacting random walks refer to a variety of models for complex large-scale disordered phenomena, motivated by questions in statistical physics, biology, economics and statistics. The interaction feature of these models produces long- memory effects which cannot be analyzed by the tools of classic probability theory.

Our project aims at developing generic tools, at understanding better its deep links with other topics, as well as to tackle new questions and applications. More specifically, we intend to work on the following topics:

• Gaussian free field (GFF), Branching random walks (BRWs) and Random interlacements • Random walks in random environment (RWRE)
• Self-interacting random walks (SIRWs)
• Particle systems, forest fires and percolation

These are fundamental models of probability, which gather essential phenomonological structure and difficulties in stochastic processes and statistical physics.

The GFF is at the nexus of most of these models. For instance it is intimately related to random interlacements through the isomorphism theorem. The level set of the GFF and random interlacements bear a key difficulty, in that their percolative structure is mean-field and not local. Important progress has been done on this, and several questions on the critical values remain open.

The BRW, which describes a randomly moving cloud of branching particles, was introduced in the context of physics and population genetics ; it is also deeply related to the 2d GFF, and consequently to quantum gravity. Indeed, it has proved to be a central tool in the understanding of the maxima of the GFF. One other very interesting direction concerns the BRW with selection, on which we have a strong expertise : several conjectures of physicists remain open.
RWRE is a fundamental model in statistical physics, describing the movement of a particle in a highly disordered medium. It has been introduced as a model of DNA chain replication and crystal growth (see Chernov and Temkin) or turbulent behavior in fluids through a Lorentz gas description (Sinai 1982). Major open questions have resisted repeated and persistent attempts to answer them, in particular invariance principles, ballisticity conditions in the non-reversible setting, and in the case of a dynamic environment. The project gathers some of the world experts in that area.

SIRWs are random processes evolving in an environment constantly modified by their own behavior, being for instance self-repelling or self-attracting. They describe a variety of phenomonological behavior and provide a good theoretical tool for learning behavior, with a link to game theory and statistical learning. This is a relatively new subject, which has recently shown deep relation with quantum field theory, Anderson localisation and the KPZ universality class. One important goal is to understand the structure of this apparently very diverse family of models.

Random walks may interact not only with their environment but also with each other, as a system of particles. This can for instance provide models for random growth, forest fires and self-organised criticality. We shall explore the connections with the (deterministic) coagulation-fragmentation equations, which often provide most useful hints for explaining properties in the random setting.

Finally, note that all these models share deep connections and a large class of technical tools. For instance techniques from RWRE, such as renewal structures and environment seen from the particle have found fruitful applications into the study of SIRWs or the propagation front of infectious models. Also, BRW is a central tool to investigate RWRE on trees.

The project will favor collaboration between french and swiss teams, already close scientifically, and will stimulate common research projects and co-supervision of doctoral theses and post-doctoral researchers. An annual conference will also be organized, alternatively by each team of the project.