# Students

**(MMath) 4th year projects 2023/24** (more information via email to Stefan Adams):

- Large deviations for random walks or Brownian motions (deviation from the law of large numbers & connection with entropy): several projects linked to MA4L3; MA3K0; MA4F7 ;MA4F7; MA4A7; MA4L2; MA482
- Scaling limits for random fields (CLT type including large deviation and moderate deviation scales)
- Large deviations for random permutations and the shape of Young tableaux - an application of MA4L3.
- Neural networks (statistical mechanics) and machine learning
- Random chaining (data science) and renormalisation (MA3K0; MA4L3)
- Wasserstein gradient flows as stochastic processes
- High-Dimensional probability - MA3K0 - concentration inequalities
- Central limit theorems for random walks
- Statistical Mechanics for Machine Learning (MA4L3)
- Continuum percolation models in dimension $d=2 $
- Large deviation for queuing and communication networks (MA4L3)
- Space-time random walks

**Two examples MMath projects from 2023:**

**Large deviations for symmetries Brownian bridges**(*large system of Brownian motions under symmetrised initial-terminal conditions; empirical path measures; Donker-Varadhan rate function*)**Continuum percolation for random Voronoi cells**(Dependent percolation; Voroni tessellations; mesoscopic limits)

**Projects can be chosen individually and can be tailored to ****specific interest (ideally discussions start in June or beginning of term or over the summer). I am going to teach MA4L3 (Large deviation theory) in term 1 and MA3K0 (High-Dimensional Probability) in term 2. Both modules can be used aa a base for a project. For MA4L3 I am going to write a new lecture book on 'Large deviation theory' for undergraduate and beginning postgraduate students which will be used as lecture for MA4L3. **

**MSc projects** (list of examples - more via discussion with Stefan Adams):

(1.) Large deviations for integrated random walks (with different applications)

(2.) Scalings limits for cycle distributions (non-standard CLTs)

(3.) Concentration inequalities - MA3K0

(4.) Gaussian free field (discrete) and its scaling to the continuous version

(5.) Permanental point processes

**Ph.D. projects** (list of examples of possible directions/questions):

(I) Renormalisation group theory & regularity structure (nonlinear SPDEs) via infinite-dimensional Laplace integral methods

(II) Large deviations for multi-dimensional random walks under pinning constraints and their multiple stochastic limits

(III) Gaussian Free Field (real and complex) and loop measures

(IV) Space-time random walks for quantum statistical mechanics

(V) Concentration inequalities for dependent random variables - applications in data science and machine learning

** **

**Current Ph.D. students:**

**Spyridion Garouniatis** (August 2021-) - *Scattering theory for many Brownian motions *

**Andreas Koller** (September 2020 -) - *Renormalisation (multi-scale) for massless models and scaling limits to Gaussian Free Fields*

**Sotirios Kotitsas** (September 2020 -) - *Nonlinear scaling limits and machine learning *

**Alberto Cassar** (September 2019 -) - *Large deviation theory analysis for integrated random walks (Laplacian) wetting/pinning models *

**Jason Ly** (September 2019 -) - *Interacting Brownian motion with symmetrised initial-terminal conditions *

**Former PhD students:**

**Shannon Horrigan** (September 2020) -*Continuum random cluster and Potts with Delaunay interactions*

**Quirin Vogel** (August 2020 ) (now Postdoc at NYU Shanghai) - *Geometric properties of random walk loop soups*

**Matthew Dickson** (September 2019) (now Postdoc at LMU Munich) - *Interacting Boson Gases and Large Deviation Principles*

**Owen Daniel** (September 2019) - *Bosonic Loop Soups and Their Occupation Fields*

**Alexander Kister** (September 2019) - *Sample path large deviations for the Laplacian model with pinning interaction in (1 + 1)-dimension*

**Michael Eyers** (January 2015) - *On Delaunay Random Cluster Models*

**William Nollett** (December 2013) - *Phase transitions and the random-cluster representation for Delaunay Potts models with geometry-dependent interactions*