# ST417: Topics in Applied Probability

##### ∗∗∗ Please note that this module will not be running in the 2018/19 academic year. ∗∗∗

##### Lecturer(s): Dr Nikos Zygouras

*Important: This module is only available to final year (4) integrated Masters students and MSc students. Please note that Topics in Applied Probability will only be taught if 5 students or more register for the module.*

**Lecturer(s):**

**Prerequisites: **Being comfortable with stochastic processes, law of large numbers, central limit theorem.

**Aims: **This module will cover several topics chosen from modern applied probability. The topics will be selected to demonstrate how probability theory can be used to study various phenomena in the real world. Examples might include random graphs, spatial point processes, branching processes and interacting particle systems.

**Commitment:** 3 lectures/week. This module runs in Term 2.

**Content:** In the academic year 2017-2018 the topic will be “ Random growth and random matrices”.

The famous Kardar-Parisi-Zhang (KPZ) universality states that a large class of randomly growing models

have fluctuations governed by a nonlinear stochastic partial differential equation (the KPZ equation) and fall outside the scope of the classical central limit theory. For example, the outer boundary of a colony of bacteria growing for time will exhibit fluctuations of order $t^{1/3}$, rather than $t^{1/2}$, as manifested by the central limit theorem. More surprisingly, the limiting distributions are different than gaussian and agree with distributions arising in a seemingly very different setting, that of eigenvalues of random matrices.

Both areas have witnessed remarkable progress in recent years. In random matrix theory the works of

the groups of Tao-Vu and Erdos-Yau have resolved the longstanding universality conjecture (ie limiting

eigenvalues statistics do not depend on the distribution of the matrix elements). In the field of KPZ significant

progress has been recently made in analysing the underlying combinatorial and integrable structure, explaining

(to some extend but not fully) the links to random matrix theory.

The course will aim to familiarise students with the forefronts of some fast developing research areas and equip them with a wide range of tools that can find applications in various settings beyond the focus topics.

The course will be ideal for students who want to pursue or are pursuing doctoral studies in probability but can

also be of interest to students with interests outside probability (in the second case the students are encouraged to

consult with the lecturer).

**Objectives:** By the end of the course, the student will:

- Familiarise with rapidly developing areas of modern probability.

- Understand and be able to use key methods and concepts with wide applicability.

**References:**

1) “Topics in Random Matrix Theory” by Terence Tao

2) “ Log-gases and random matrices” by P. Forrester

3) “Dynamical approach to random matrix theory” by L. Erdos and H.T. Yau

4) "A pedestrian's view on interacting particle systems, KPZ universality, and random matrices” , by T. Kriecherbauer and J. Krug

**Assessment: **100% by 2hr exam.

You may also wish to see:

ST417: Topics in Applied Probability

(restricted access)