Random Matrix Theory, Quantum Chaos and Statistical Mechanics 

Random matrices are often used to study the statistical properties of systems whose detailed mathematical description is either not known or too complicated to allow any kind of successful approach. It is a remarkable fact that predictions made using random matrix theory have turned out to be accurate in a wide range of fields: statistical mechanics, quantum chaos, nuclear physics, quantum transport, number theory, combinatorics, wireless telecommunications, quantum field theory and probability, to name only few examples. My research has focused mainly on applications of random matrices to quantum transport, quantum chaos, statistical mechanics and on the universality properties of the statistics of the eigenvalues.

PhD Projects

I am open to supervise projects in most current areas of research in Random Matrix Theory. Interested students should contact me by email for enquiries, f.mezzadri@bristol.ac.uk. In particular, my interests include the following topics.

Two-Dimensional One Component Plasma and Non-Hermitian Random Matrices

The joint probability density function of the eigenvalues of Non-Hermitian Matrices has the same form of the Boltzmann factor of a two-dimensional plasma of Coulomb charges 2D-OCP. This statistical mechanics fluid model has appeared in several areas of physics and mathematics. Indeed, the logarithmic repulsion of the charges occurs as interaction between vortices and dislocations in systems such as superconductors, superfluids, rotating Bose-Einstein condensates. There is also an analogy between the 2D-OCP and the Laughlin trial wave function in the theory of fractional quantum Hall effect. We can apply techniques from Random Matrix Theory to gain a better understanding of the behaviour of these systems. A PhD project can be chosen among many open problems in this areas. See Cunden, F.D., Mezzadri, F. & Vivo, P. “Large deviations of radial statistics in the two-dimensional one-component plasma.” J Stat Phys (2016) 164: 1062. doi:10.1007/s10955-016-1577-x

Random Matrix Theory and Quantum Transport

Quantum transport in disordered mesoscopic conductors have important applications in modern technology, where it has become increasingly important to miniaturise components of electronic devices. In the 1980s it was discovered that the statistical fluctuations of the conductance in disordered quasi one-dimensional wires are universal, which means that within certain limits they are independent of the size of the sample and strength of the disorder. Soon afterwards it was realised that Random Matrix Theory could provide the mathematical framework to develop a statistical theory of quantum transport that would account of the universality of the fluctuations of the electric current. Recently a lot progress in this area has been achieved exploiting the link between Random Matrix Theory and integrable systems. There are still many open questions that can be answered using these ideas and that can be chosen as a PhD project. See. Mezzadri, F. and Simm, N. J. “Tau-Function Theory of Quantum Chaotic Transport with beta=1,2,4.” Commun. Math. Phys. (2013) 324: 465. doi:10.1007/s00220-013-1813-z