Description
Matrix models are statistical systems whose degrees of freedom are matrices. All the observable quantities of a matrix model are computed by integrals over ensembles of matrices, with an appropriate measure. Alternatively, matrix models can be thought of as toy models of quantum field theories: they are quantum gauge theories in zero spacetime dimensions.
Unlike gauge theories in higher dimensions, matrix models are often solvable: the relevant integrals can be computed exactly, for matrices of arbitrarily large size. The solution can be obtained by a variety of methods, all of which bypass brute force integration.
Despite being very simple to formulate, matrix models have many uses in theoretical physics and mathematics, which range from gauge theory, condensed matter physics and quantum gravity to combinatorics, number theory and algebraic geometry. All of this can be understood by studying simple integrals over matrices.
The aim of this project is to investigate some of the methods of solution of matrix models (large N expansion, orthogonal polynomials, loop equations) and explore some of their physical and mathematical applications.
Pre- and co-requisites
2H Mathematical Physics (or equivalent)
3H Quantum Mechanics.
The project does not require any prior knowledge of quantum field theory or statistical mechanics, but you should be familiar with linear algebra and prepared to acquire a variety of computational skills along the way.
Resources
For some context:
Reading material:
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