Copy of Probability and statistical physics: interacting particle systems, growth models and percolation
Lead Research Organisation:
University of Oxford
Department Name: Statistics
Abstract
Many phenomena can be described in terms of the growth of a cluster which evolves according to random interactions at its boundary: the growth of a crystal, the spread of a biological population across an area, the spread of a rumour in a group of acquaintances or of a virus on the internet.Consider the following simple example of a growth model. A set of sites is organised according to a square grid, and initially a single site is occupied. Once any neighbour of a site is occupied, a random amount of time passes before that site becomes occupied itself.The cluster of occupied sites grows over time. There are many different questions of interest, for example: as the cluster becomes large, does it have a fairly predictable shape? How smooth is the boundary of the cluster? What is the correlation between the times at which two different sites join the cluster, and how does this depend on the distance between them?Extremely precise answers to questions of this kind are known for certain very specific growth models. However, it is widely believed that many aspects of the behaviour should not depend strongly on the exact nature of the growth rules. A major aim of this project is to justify this belief by finding rigorous mathematical proofs of universality properties, using the tools of probability theory.There are close relations to so-called interacting particle systems . For example, consider a system in which particles are located at positions on a line, given by numbers in the set {...,-3,-2,-1,0,1,2,3,...}. Particles try to jump to their right at random times; the jump succeeds if the position to the right is not already occupied by another particle. This very simple system turns out to possess rich behaviour, and is widely used in statistical physics as a basic model of non-reversible flow. Certain interacting particle systems of this type are in fact equivalent to growth models (for example, one can imagine that the point in the growth model with co-ordinates (n,k) joins the occupied cluster at the time when particle n jumps to position k in the particle system). The same universality questions arise: to what extent do quantities of interest (such as average rates of flow, or correlations between the positions of different particles) depend on the precise nature of the randomness of the jumps and the rules of the local interaction between particles?A second strand of the project concerns combinatorial optimisation or constraint satisfaction problems. A huge variety of problems in computer science can be represented in an abstract way by simple formulae involving variables which take the values True or False, joined together by the operators AND and OR. A formula is said to be satisfiable if there is an assignment of the values True or False to the variables so that the overall formula has the value True. An extremely important question concerns computational complexity: how long does it take for an algorithm to find such a satisfying assignment, or to prove that none exists? For certain classes of problem, it is believed that the set of satisfying assignments for a given formula tends to have clustering properties, rather than being more or less evenly spread out across the space of all assignments, and that this phenomenon creates problems with high complexity.In this project I will relate such clustering properties to the behaviour of interacting particle systems whose particles take positions in a tree-like structure, and in particular to the set of possible equilibrium states for such a system. This can be seen as part of a much broader programme at the interface between mathematics, physics and computer science, aiming to give a sound mathematical footing to a class of very powerful but non-rigorous tools which originate in statistical physics.
Organisations
People |
ORCID iD |
James Martin (Principal Investigator) |
Publications
Angel O
(2013)
Avoidance Coupling
in Electronic Communications in Probability
Basu R
(2016)
Trapping games on random boards
in The Annals of Applied Probability
Basu Riddhipratim
(2015)
Games on Random Boards
in arXiv e-prints
Ferrari P
(2009)
A phase transition for competition interfaces
in The Annals of Applied Probability
Ferrari P
(2009)
Collision probabilities in the rarefaction fan of asymmetric exclusion processes
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Ferrari P
(2009)
Multiclass Hammersley-Aldous-Diaconis process and multiclass-customer queues
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Foss S
(2014)
Long-range last-passage percolation on the line
in The Annals of Applied Probability
Holroyd A
(2014)
Stochastic domination and comb percolation
in Electronic Journal of Probability
Holroyd A
(2018)
Percolation games, probabilistic cellular automata, and the hard-core model
in Probability Theory and Related Fields
Description | The work of this project has extended and enhanced our understanding of a range of phenomena arising in areas including probability theory, statistical physics and theoretical computer science. The phenomena are related to processes evolving in space, such as models of random growth, percolation, or coagulation and fragmentation; also to problems involving combinatorial optimisation. Together with collaborators I have identified new scaling limits describing the behaviour of large systems, and new ways to give exact descriptions of the equilibrium states of spatial models involving the interaction of large numbers of particles of different types. We have also identified novel areas in which related techniques can be applied, including to problems involving spatial tesselations (relevant to mobile communications) and scheduling problems (relevant in a variety of fields including for example supply chains and communication networks). We have also developed new theoretical tools which are already being adopted and adapted by other researchers for the analysis of models related to percolation and random growth. |
Exploitation Route | Several of the theoretical tools developed are already used by other researchers working on percolation and random spatial processes. Probabilistic approaches developed in the course of the project are also being taken up and extended in a fascinating way by other researchers whose expertise is in algebraic combinatorics. This leads to unanticipated connections between random spatial models and algebraic structures such as walks in permutation groups. Many collaborative outputs of the project are currently under submission, in draft, or in the course of being written up - as a result many further publications representing outcomes of the project will appear over time. Nonetheless we have taken the opportunity to disseminate key ideas at international workshops and conferences, resulting in several subsequent and continuing collaborative projects. |
Sectors | Digital/Communication/Information Technologies (including Software),Other |
URL | http://www.stats.ox.ac.uk/~martin/papers.html |
Description | The project lies in an extremely rapidly evolving area at the intersection of probability theory, statistical physics and theoretical computer science. The findings of the project enhance the understanding of a range of models relating to random spatial processes, processes evolving on networks, and problems of combinatorial optimisation. |
First Year Of Impact | 2013 |
Sector | Digital/Communication/Information Technologies (including Software),Other |
Description | ALEA Network workshop in Bordeaux |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Postgraduate students |
Results and Impact | Research-level course on "Percolation games, branching processes, and probabilistic cellular automata" at ALEA Network workshop in Bordeaux, aimed at postgraduate students, as well as postdoctoral researchers and others. |
Year(s) Of Engagement Activity | 2015 |
URL | http://www.labri.fr/perso/marckert/Alea-Network_LaBRI_2015.html |
Description | Queueing systems and particle models |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Type Of Presentation | keynote/invited speaker |
Geographic Reach | International |
Primary Audience | Professional Practitioners |
Results and Impact | Research-level course given at the CIMPA graduate school "Stochastic dynamics of particles and networks", Mar del Plata (Argentina), 19-30 November 2012 http://mate.dm.uba.ar/~probab/cimpa/. A particularly large proportion of the audience were graduate students from South American countries. The activity has strengthened research links between the UK and South American communities in the area. Several of those attending the workshop have subsequently applied for postdoctoral-level positions in the UK. |
Year(s) Of Engagement Activity | 2012 |
URL | http://mate.dm.uba.ar/~probab/cimpa/ |