KL7015 - Complex and Random Systems

What will I learn on this module?

You will learn about a range of appropriate statistical techniques that are used to predict and analyse complex systems modelled by random matrices. You will be introduced to the generalisation of probability theory for multivariate calculus, the analysis of the most common ensembles (Gaussian Orthogonal and Unitary Ensembles, the Circular Ensembles) and methods for using these tools efficiently in numerical simulations.

Outline Syllabus
– Review of linear algebra and probability theory
– Numerical techniques to generate and analyse random matrices
– The Circular Unitary Ensemble (CUE): definition, spacing distribution, eigenvalues correlation functions
– The Circular Orthogonal Ensemble (COE)
– The Gaussian Ensembles: unitary, orthogonal, symplectic
– Orthogonal polynomial techniques (large N limit and universality)

Depending on the time the extrema statistics (Tracy-Widom distribution) will be derived as it can be found in numerous applications (combinatorics, biology)

How will I learn on this module?

You will learn through a series of lectures, seminars and problem-solving/computer-based workshops which include classroom discussions and presentations. Lectures allow you to witness the development of the theory of random matrices to provide with predictions of certain observables in complex systems, and understand how to apply the required techniques coming from several areas in mathematics.
Seminar classes and problem-solving/computer-based workshops will be scheduled weekly to allow exploration of the theoretical background to the techniques covered in the lectures.

Formative feedback is available weekly in the classes as you get to grips with new techniques and solve problems. In addition, we operate an open door policy where you can meet with your module tutor to seek further advice or help if required.

How will I be supported academically on this module?

Direct contact with the teaching team during the lectures and seminars will involve participation in both general class discussions as well as one to one discussions during the seminars. This gives you a chance to get immediate feedback pertinent to your particular needs in this session. Further feedback and discussion with the teaching team are also available at any time through our open door policy. In addition, all teaching materials, selected computer programmes and supplementary material (such as interesting articles) are available through the e-learning portal.

What will I be expected to read on this module?

All modules at Northumbria include a range of reading materials that students are expected to engage with. Online reading lists (provided after enrolment) give you access to your reading material for your modules. The Library works in partnership with your module tutors to ensure you have access to the material that you need.

What will I be expected to achieve?

Knowledge & Understanding:
1. Use specific random matrix ensembles to represent a large systems with complex interaction between their components
2. Evaluate observables and analyse statistical distributions

Intellectual / Professional skills & abilities:
3. Construct suitable models based on qualitative properties of systems and data sets.
4. Classify and assess occurrence of possible scenarios.

Personal Values Attributes (Global / Cultural awareness, Ethics, Curiosity) (PVA):
5. Critically appraise suitability of models and validation of assumptions.

How will I be assessed?

SUMMATIVE
1. Coursework: 50% - 1,3
(Assignment with set questions and problems - wordcount: max 1000 words + derivations + codes + graphs and plots)


2. Group project: 50% - 1,2,4,5
(Group work: 20min presentation with electronic slides + 10min questions/answers and discussion)
FORMATIVE
Formative assessment will be available on a weekly basis in the seminars through normal lecturer-student interactions and discussions around the seminar questions, allowing them to extend, consolidate and evaluate their knowledge.

Formative feedback will be provided on student work and errors in understanding will be addressed reactively using individual discussion. Solutions for seminar tasks will be provided after the students have attempted the questions, allowing students to receive feedback on the correctness of their solutions and to seek help if matters are still not clear.

Pre-requisite(s)

None

Co-requisite(s)

None

Module abstract

In “Random and complex systems” you will learn techniques to model statistical properties of systems with a large number of components using random matrix ensembles. Random matrices, originally introduced in high energy physics to describe properties of heavy atoms, have become a powerful tool for modelling a variety of systems with random and complex features. Applications range from physics (e.g. condensed matter to quantum gravity), mathematics (number theory, combinatorics), complexity science (traffic, communication, biological networks) and economics (stock and option pricing in financial markets).

You will learn through a combination of lectures andseminars. The lectures give a formal introduction to the theoretical aspects while the workshops enables one to deepen the knowledge by applying the theory to problems to more practical questions arising from physics, and complexity science. The workshops will be an opportunity to span the wide range of applications of random matrices, thus also strengthening your transferable skills and employability.

You will be assessed through a coursework and a group project presentation

Course info

UCAS Code G101

Credits 20

Level of Study Undergraduate

Mode of Study 4 years Full Time or 5 years with a placement (sandwich)/study abroad

School Engineering, Physics and Mathematics

Location City Campus, Northumbria University

City Newcastle

Start September 2027

Fee Information

Module Information

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