The HyLEF project (Hydrodynamic Limits and Equilibrium Fluctuations: universality from stochastic systems) is one of the projects funded by the European Research Council(ERC) in the 2016 edition of the ERC Starting Grants. This is the first ERC grant awarded in Portugal in the field of mathematics and is headed by Patrícia Gonçalves, an associate professor at the Mathematics Department of Instituto Superior Técnico of theUniversity of Lisbon. The five-year grant is worth nearly €1.2m and started on 1 December 2016, with the budget funding a team composed of three postdocs (two years each), two PhD students (four years each) and two master’s students (one year each). This will be the first team in Portugal working on the interacting particle systems field of mathematics. There is also budget for organising two conferences over the five years and to invite external collaborators to work with the team in Portugal. The core of the project is related to the search for ‘universality’ from microscopic stochastic systems.
What is universality? To exemplify what is meant by universality, in Fig. 1 we see a pattern of ice particles formed on a windscreen. The ice particles fall randomly and form a pattern that can be seen in other, uncorrelated situations, such as coffee ring effects, bacterial growth (e.g. E-coli), in the wake of a flame, tumour growth, etc. Many different physical systems, when analysed from a mathematical point of view, show identical patterns of growth. This slightly mysterious tendency for very different things to behave in very similar ways is the essence of universality.
The shape of these patterns is the subject of this project and it is related to universality, a very active area of research in both mathematics and physics. One can think of the ice particles as blocks that fall independently but whenever one falling block touches another, even only from the side, it immediately sticks into place. This is a kind of random deposition of blocks that leads to the pattern that we have in Fig. 1.
There are different types of random deposition of blocks, for example allowing the blocks to fall independently and in parallel above each site. In this case there is no stickiness. These two different types of random deposition give rise to two different universality classes.
Mathematically, this is a way of characterising systems that somehow share the same properties. We have mentioned two universality classes: the random deposition is in what is called the ‘Gaussian universality class’ and the sticky deposition is in the ‘Kardar-Parisi-Zhang universality class’. This project focuses on the latter.
In 1986, physicists Kardar, Parisi and Zhang proposed a phenomenological model for the evolution of a random growing interface given by what is called the ‘KPZ equation’. This project aims at exploring the universality of the KPZ equation from underlying microscopic stochastic dynamics. The latter consists of a collection of particles which move after a random time according to some fixed probability transition rate. Mathematically speaking, this collection consists of a Markov process whose dynamics conserve some quantity of interest. Some of the questions stated in this project are: what are the macroscopic laws governing the evolution of the conserved quantities of microscopic stochastic dynamics? What are the universality classes to which the models, with certain general features, belong? What is the relation between these classes? Are they linked by some parameter given on the underlying microscopic stochastic dynamics? The goal of the project is to answer these
questions but with the focus on universality.