About the Project
The University of Bath is inviting applications for the following PhD project commencing in September 2024.
We offer a project of PhD thesis in Probability. The student would work with Dr Christoforos Panagiotis, at the University of Bath. The goal of the thesis will be to prove sharpness results and study the critical behaviour of correlated percolation models.
Lattice spin models are fundamental in statistical physics, representing archetypal examples of systems undergoing phase transitions. These models involve spins located on the vertices of a graph, interacting with their neighbours, and their values are constrained according to a model-dependant probability measure. While some of these models, such as the Ising model, have undergone extensive study, there is an ongoing interest in developing robust techniques that can be applied broadly. This includes models that can be realised as Ising models in random environments, with a primary example being the Blume-Capel model.
Percolation representations allow to express spin correlations in terms of connection probabilities between vertices by associating spin configurations with percolation clusters. Adapting recent techniques developed by Duminil-Copin, Raoufi and Tassion could help address one of the major conjectures, which states that the phase transition is sharp. In percolation terms, this translates to demonstrating that the size of the largest cluster inside a large box changes drastically from logarithmic to linear.
Understanding the system’s behaviour at its critical point can be more challenging, but techniques from percolation theory developed by Dumnil-Copin and Tassion have been particularly effective in dimension 2, where due to planar duality, the asymptotic behaviour of crossing probabilities can give important information on the critical exponents of the model. In higher dimensions, spin models are expected to behave like Gaussian fields in large scales, in contrast to what happens in lower dimensions. Random current representations and random walk expansions, such as the one obtained recently by Gunaratnam, Panagiotis, Panis and Severo, provide a random walk perspective and allow to capture this behaviour by translating it into the problem of whether random walks in high dimensions do not intersect.
The project will involve learning a range of techniques from probability, combinatorics, and geometry, and the student will focus on studying percolation and random walks.
The full advertisement can be found here: https://www.findaphd.com/phds/project/stochastic-geometry-and-applications-to-lattice-spin-models/?p162107.