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## Probability

### Organisation profile

Probability theory is concerned with evaluation of chances where the central subjects include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion.

Classical areas of research in probability are Limit Theorems and Large Deviations. It goes back to Jacob Bernoulli in 1713 that, if we repeat independently some stochastic experiment many times, then the frequency of some event stabilises in a long run, this law is known as the law of large numbers. Modern probability theory provides further developments in this direction, among them the central limit theorem which is one of the great results of mathematics. It explains the ubiquitous occurrence of the normal distribution in nature.

Another classical area of researh concerns Markov Processes which goes back to the early 20th century. It is an extension of independent random sequences where predictions can be made regarding future outcomes based solely on the present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. Research has reported the application and usefulness of Markov chains in a wide range of topics such as physics, chemistry, biology, medicine, music, game theory and sports.

Among other areas of research in our department is one related to random growth processes which describe growing objects that evolve over time according to some underlying random structure. These processes occure often in nature. Examples include tumoral growth, lightning strikes and mineral aggregation. We often want to study this growth in order to  understand the underlying natural process.

An active research is also conducted in the area of Noncommutative Probability. In particular, Free Probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products. This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras.

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• Published

## Dissipative particle systems on expanders

Haslegrave, J. & Keevash, P., 25/04/2024, Arxiv.

Research output: Working paper

• Published

## Scaling limits for planar aggregation with subcritical fluctuations

Norris, J., Silvestri, V. & Turner, A., 28/02/2023, In: Probability Theory and Related Fields. 185, 1-2, p. 185-250 66 p.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

• Published

## Crux and Long Cycles in Graphs

Haslegrave, J., Hu, J., Kim, J., Liu, H., Luan, B. & Wang, G., 31/12/2022, In: SIAM Journal on Discrete Mathematics. 36, 4, p. 2942-2958 17 p., 4.

Research output: Contribution to Journal/MagazineJournal articlepeer-review

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• ## Manchester Metropolitan University

Activity: Visiting an external institution typesVisiting an external academic institution

• ## Mediterranean Journal of Mathematics (Journal)

Activity: Publication peer-review and editorial work typesPublication peer-review

• ## Infinite Dimensional Analysis, Quantum Probability and Related Topics (Journal)

Activity: Publication peer-review and editorial work typesEditorial activity