Graphical representations of classical and quantum spin systems

Summary

This project will explore connections between models of Statistical physics (The Bose gas, the quantum and classical (anti-)ferromagnet, the Ising model), related probabilistic models (the random path model, the random interchange/stirring model, the random current model) and some "complex" spin systems (that is - spin systems where spins take complex values).

The aim will be to understand properties of these models such as phase transitions, percolation, decay of correlations by using the connections between the three familes of model and hence learn more about the models than would be possible by looking at only one of the three families.

Many models from Statistical Physics, such as the Bose gas, (quantum and classical) ferromagnet/anti-ferromagnet, and the Ising model, have probabilistic representations in terms of one-dimensional object (Brownian or random walk paths, loops, dimers….) living in the same space as the physical model. This space could be a graph such as a regular lattice, or a subset of the familiar three-dimensional reals. It has recently been noticed that they can also be representation as a different kind of spin system where spins take complex values.

These representations have many attractive properties for Mathematicians. They are often quite beautiful probabilistic models for study in their own right but also give the opportunity to learn about important models from Physics with a new array of tools. They do however come with challenges, many of these models have complicated correlations that means there is no (or very little) independence in the model. This difficulty can sometimes be overcome, but even when it cannot there are tools available to us.

This project will explore these models both as probabilistic objects isolated from Physics and as physical models. Questions of percolation and other phase transitions in these models will be explored, as well as decay of correlations. This will be achieved by using the complex spin and probabilistic representations which allows us to switch between different techniques and approaches.

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