## Postgraduate research opportunity

# Random Permutations and Integer Partitions

**Value**: This project is open to self-funded students and is eligible for funding from the School of Mathematics Scholarships, EPSRC Doctoral Training Partnerships, and Leeds Doctoral Scholarships.**Number of awards**: Variable**Deadline**: Applications accepted all year round**Key benefits**: Our research focuses on the study and modelling of systems and processes featured by uncertainty and/or complexity, using advanced theoretical, simulation and numerical methods. It covers a vast variety of modern topics both in probability (including theory of random processes and stochastic analysis) and in a wide range of applications in mathematical and other sciences, spanning from nonlinear dynamical systems and mathematical physics through mathematical biology and complexity theory to mathematical finance and economics.

### Supervisor(s)

Contact Dr Leonid Bogachev to discuss this project further informally.

### Project description

You will be based within a strong research group in Probability, Stochastic Modelling & Financial Mathematics.

Permutations and integer partitions are the basic combinatorial structures that appear in numerous areas of mathematics and its applications — from number theory, algebra and topology to quantum physics, statistics, population genetics, IT & cryptology (e.g., Alan Turing used the theory of permutations to break the Enigma code during World War II). This classic research topic dates back to Euler, Cauchy, Cayley, Lagrange, Hardy and Ramanujan. The modern statistical approach is to treat these structures as a random ensemble endowed with a suitable probability measure.

The uniform (equiprobable) case is well understood but more interesting models (e.g., with certain weights on the components) are mathematically more challenging. The main thrust of this PhD project is to tackle open and emerging problems about asymptotic properties of "typical" structures of big size. The focus will be on macroscopic features of the random structure, such as its limit shape.

It is also important to study extreme values, in particular the possible emergence of a giant component which may shed light on the Bose–Einstein condensation of quantum gas, predicted in 1924 but observed only recently (Nobel Prize in Physics 2001). A related direction of research is the exploration of a deep connection with different quantum statistics; specifically, the ensemble of uniform integer partitions may be interpreted as the ideal gas of bosons (in two dimensions), whereas partitions with distinct parts correspond to fermions. In this context, an intriguing problem is to construct suitable partition classes to model the so-called anyons obeying fractional quantum statistics (also in 2D!). Furthermore, an adventurous idea may be to look for suitable partition models to mimic the unusual properties of graphene (Nobel Prize in Physics 2010), a newly discovered 2D quantum structure with certain hidden symmetries.

*REFERENCES*

*[1] Arratia, R., Barbour, A.D. and Tavaré, S. Logarithmic Combinatorial Structures: a Probabilistic Approach. European Math. Soc., Zürich, 2003. (doi:10.4171/000)*

*[2] Bogachev, L.V. Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts. Random Struct. Algorithms, 47 (2015), 227–266. (doi:10.1002/rsa.20540)*

*[3] Bogachev, L.V. Limit shape of random convex polygonal lines: Even more universality. J. Comb. Theory A, 127 (2014), 353–399. (doi:10.1016/j.jcta.2014.07.005)*

*[4] Bogachev, L.V. and Zeindler, D. Asymptotic statistics of cycles in surrogate-spatial permutations. Comm. Math. Phys., 334 (2015), 39–116. (doi:10.1007/s00220-014-2110-1)*

*[5] Lerda, A. Anyons: Quantum Mechanics of Particles with Fractional Statistics. Springer, Berlin, 1992.*

*[6] Vershik, A.M. Asymptotic combinatorics and algebraic analysis. In: Proc. Intern.Congress Math. 1994, vol. 2. Birkhäuser, Basel, 1995, pp. 1384–1394. (www.mathunion.org/ICM/ICM1994.2/Main/icm1994.2.1384.1394.ocr.pdf)*

### Entry requirements

Applications are invited from candidates with or expecting a minimum of a UK upper second class honours degree (2:1), and/or a Master's degree in a relevant mathematics subject such as (but not limited to) probability; combinatorics; mathematical statistics; analysis; physics.### How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is the ‘Random Permutations and Integer Partitions’ as well as Dr Leonid Bogachev as your proposed supervisor.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

*We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.*