# Students

**4th year projects 2016** (more information via email to Stefan Adams):

(1.) Large deviations for random walks or Brownian motions (deviation from law of large numbers & connection with entropy): several projects linked to MA4L3.

(2.) Large deviations for random integer sequences and their minimisers - an application from MA4L3.

(3.) Large deviations for random partitions and teh scaled shape limits of Young tableaux - an application from MA4L3.

(4.) Isomorphism theorems for loop measures (on a graph or in $ \R^d$ defined via random walks where a loop is a closed path (finite time horizon)).

(5.) Percolation models (Delaunay graphs, Voronoi graphs, triangle lattice; Euclidean plane) and their critical regime.

(6.) Neural networks

**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-standrad CLTs)

(3.) Percolation for interacting Voronoi fields in the Euclidean plane and conformal invariance

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

**PhD projects** (list of examples of possible directions/questions):

(I) Renormalisation group theory via infinite dimensional Laplace integral methods

(II) Large deviations for multi-dimensional random walks under pinning constraints and their percolation behaviour

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

(IV) Continuum percolation for Delaunay models in $ \R^2 $ and other random graph model in two dimensions

(V) Breaking of continuous symmetries - mathematical approach to crystallisation

**Current PhD students:**

Matthew Dickson (MASDOC) (September 2015- ) - *Bose-Einstein condensation phenomena and scaling limits for a class of mean-field models*

Quirin Vogel (MASDOC) (September 2016 - ) - Space-time random walks, scaling limits and Bosonic loop measures

**Former PhD students:**

Owen Daniel (September 2011-September 2015) - *Bosonic Loop Soups and Their Occupation Fields*

Alexander Kister (September 2015) - *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*