Random Permutations and Integer Partitions with Structural Constraints

Summary

Permutations and integer partitions are the basic combinatorial structures that appear in numerous areas of mathematics and its applications. Modern statistical approach is to treat such structures as a random ensemble endowed with a suitable probability measure. Structures with certain constraints on their components are mathematically challenging. The main thrust of this PhD project is to study properties of big structures, focusing on macroscopic features such as limit shape.

Permutations and integer partitions 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 such combinatorial 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, especially under certain structural constraints on the consitituent components. 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

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3. Bogachev, L.V. Limit shape of random convex polygonal lines: Even more universality. Journal of Combinatorial Theory A, 127 (2014), 353–399. (doi:10.1016/j.jcta.2014.07.005)
4. Bogachev, L.V. and Yakubovich, Yu.V. Limit shape of minimal difference partitions and fractional statistics. Communications in Mathematical Physics, 373 (2020),1085–1131. (doi:10.1007/s00220-019-03513-5)
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6. Lerda, A. Anyons: Quantum Mechanics of Particles with Fractional Statistics. Springer, Berlin, 1992.
7. Vershik, A.M. Asymptotic combinatorics and algebraic analysis. In: Proceedings of the International Congress of Mathematicians (August 3–11, 1994, Zürich, Switzerland), Vol. 2. Birkhäuser, Basel, 1995, pp. 1384–1394. (doi:10.1007/978-3-0348-9078-6_133)

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline.

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid.