Stochastic interacting systems: connections, fluctuations and applications

Many classical results in probability theory consider independent (or weakly dependent) and often identically distributed random variables. The most fundamental questions regarding convergence of the sample average (Law of Large Numbers) and the fluctuations thereof (Central Limit Theorem) are well understood in these cases. The picture becomes very different when independence is lost.

Several models have been invented to help our understanding of observations in diverse phenomena such as condensed matter and statistical physics, molecular, cell-level and population biology, geography, sociology and also engineering. A common feature of the processes considered here is the loss of independence (or weak dependence). Atomic transport processes in physical systems interact with each other, as well as molecules do when progressing in small cellular channels, or blood cells in narrow vessels of the body. Engineering sees similar interactions between participants of traffic flow that is happening on our streets. Chemotaxis and other autonomous motion of biology depend on the local environment of the moving cell, making its behaviour correlated to any earlier segment of the path that saw the same environment. Infections in a population or forest fires among trees advance as a randomly growing surface where the speed of growth depends on the shape of the very same surface in a local neighbourhood. Avalanches are modeled as self-reinforcing interacting random motion of blocks of snow, while changes in one stretch of a riverbed strongly influence those in other bits. Voting opinions and rumour spreading is obviously a very cross-correlated process of sociology with lots of spatial interactions. Data-transmission systems use queues that interact in complicated ways via routing algorithms. In all of the above examples any attempts to build a stochastic model of observations quickly lead to stochastically dependent sequences of random variables.

The temporal and/or spatial dependence in many of these processes makes it impossible to apply the classical methods. In some instances new ideas can be invented to prove the Law of Large Numbers and Central Limit Theorem behaviour. In some other cases the scaling of the Central Limit Theorem and the Normal distribution being the universal limit will simply not be valid anymore. New, still very universal scaling orders and limit distributions emerge, characteristic to general classes of interacting processes but essentially different from the usual independent picture. An example where several groundbreaking results have been achieved in recent years is the so-called Kardar-Parisi-Zhang equation with its characteristic time^{1/3} scaling and Tracy-Widom limit distributions.

Mathematical research in these areas thus require essentially new ideas that often strongly interact with various fields of mathematics (functional analysis, algebra, combinatorics, dynamical systems among others) and physics. Our research aims at fundamental questions of constructions, stationary behaviour, fluctuations and scaling limits in some of the above models. More specifically, we investigate:
- random walks both in fixed and dynamically changing random environments, where an otherwise simple random motion changes its behaviour depending on its position;
- interacting particle systems, where many, otherwise simple motions interact with each other.
Each of our research questions concerns cases that are far from the classical well-established scenarios. To come up with results we apply original probabilistic ideas and tools from other fields of mathematics. The behaviour we can demonstrate in these systems is new, and greatly improves our general understanding in other sciences which use these models. Our work also represents valuable contributions to mathematics because of interactions with other areas and because of the variety of ideas that one necessarily invents in the lack of traditional tools.

Please see the Academic beneficiaries section for various details regarding the academic impact of our research. It is important to note that successful completion of our research programme will also have a significant positive impact on the UK's competitiveness in comparison to other leading research centers in the field (such as the Massachusetts Institute of Technology, Institut Henri Poincare, ETH Zurich, etc.).

Basic mathematical research often finds its way to everyday applications in unpredictable and surprising ways, and we expect our results to contribute in engineering, geographical sciences and sociology in the future. On more predictable grounds, one of the research packages in this proposal will directly impact the geomorphology community, see the details in the Case for Support document. Further, the PI regularly publicises some of his earlier results related to this proposal at university open days, promoting interest in mathematics and probability theory therein. He has an excellent and wide teaching record across all levels of university teaching, and this has hugely benefited from active mathematical research. Continued support of the PI's research activities thus benefits a large group of prospective and current students in higher education.