Constrained random geometries: phase boundary fluctuation and sub-ballistic motion

As a mathematician working in the field of statistical mechanics, I am pursuing research problems that concern mathematical models that exemplify some important aspect of the physical world. The aim is find the simplest viable description of the typical large scale behaviour of such systems. For example, there are huge number of air molecules in a room, so that, microscopically, the behaviour of the air in the room is described by all of the positions and velocities of those molecules. This is a vast amount of information. In the large, however, what effectively describes the air is a few parameters, such as the temperature and the pressure, that are specified by averages of particles over small regions of space (that nonetheless contain very many particles). In the rigorous theory of statistical mechanics, we choose a suitable mathematical model of a physical system, and prove how the behaviour of such macroscopic quantities as temperature and pressure arises from the microscopic structure of the system.In this proposal, I am undertaking three related research projects, each of which reflects in some way this theme:I. Phase boundary fluctuation. If oil is injected into still water, it forms into a droplet that makes the total surface tension at the boundary as small as possible. On a finer scale, however, the boundary between the two substances may be random. In a recent series of papers, I have investigated, for a natural mathematical model of two such substances, the geometry of this random boundary. I am proposing to investigate what is universal about this random fluctuation: that is, which elements of this behaviour are shared with a diverse range of other systems. II. Trapping in disordered media.If a charged particle in an electric field moves in an environment populated by occasional obstacles, its progress is liable to be frustrated by traps formed by the obstacles. What is the geometry of the traps that waylay the particle, and to what degree do these traps slow down the walk? Alexander Fribergh and I are carrying out an extensive investigation of a mathematical model of this problem, in which a walker jumps generally in a preferred direction, but makes other random moves as well, on a grid in which some edges are impassible.III. Spatial random permutations.At very low temperatures, helium condenses into a remarkable substance that flows with extreme ease. A mathematical model of repelling random particles is naturally associated to such low-temperature gases. I am planning to investigate how these particles behave in a fashion that, while random, has large scale order, and how this order is related to the special properties of very cool gases.

The research programme will advance understanding in very challenging problems in probability theory and statistical mechanics. A deeper rigorous understanding of trapping in disordered systems will give insight into the susceptibility of materials to stress. Such an understanding of phase transitions in low-temperature gases may yield a more precise understanding of the temperature at which superfluids form, ultimately benefiting in such fields as materials' technology. I aim to make progress on some of the most challenging of those problems in probability that lie at the interface with condensed matter physics. As such, there will be impact in the development of new tools for tackling these technically demanding problems. The impact of these techniques will be felt in both mathematics and physics, where researchers will have new techniques at their disposal to tackle other major unsolved problems. These aims will entail developing a group of researchers, including several graduate students, to undertake them. Members of the group will become acquainted with a diverse range of mathematical skills and will help to develop these new tools in statistical physics. Students graduating from the vibrant research group that I aim to construct might continue to become experts at the boundary of probability theory and physics, or might work in the financial industry, inventing techniques in stochastic analysis to analyse market fluctuations, or in internet search, developing algorithms that exploit an understanding of the random structure of large-scale systems. I seek to further the collaborative aims of the proposal by expanding a graduate seminar series on spatial random permutations, that I have begun to organise in Oxford with James Martin, and by beginning another on critical two-dimensional random systems.