I’m Anas Rahman, a student at the University of Melbourne, on the verge of graduating with a Bachelor of Science (Physics) and a Concurrent Diploma of Mathematical Sciences (Pure Mathematics). I intend to start a Masters degree with the Mathematics & Statistics department at the University of Melbourne next year. My studies have mainly focused on pure mathematics, with a strong background in physics and a bit of applied mathematics. As such, my research interests lie in mathematics derived from physical problems.
Last year, I participated in the University of Melbourne’s VRS program, having worked with Nicholas Witte on some random matrix theory. In particular, we investigated the moments of the Gaussian Beta Ensembles, which have applications in statistical physics (Beta values referring to the inverse temperature of physical systems). This year, I will be continuing work with Nicholas in this field, this time looking at asymptotic behaviours exhibited by these moments.
Densities and Moments of the Beta Ensembles in Random Matrix Theory
Random matrix theory has provided theoretical paradigms in many applications including the distribution of mutation fitness effects across species, principal component analysis, wireless communication and antenna networks, queuing models and quantum transport in mesoscopic systems just to name a few examples. However some of the most exciting developments in this area are happening in the purely mathematical arena.There are a number of tools available to study for example the density of eigenvalues of a random matrix for general Î² (the inverse temperature) in various ensembles each with their own advantages and drawbacks. One of these is the loop equation formalism which is well suited to the task of computing the large rank or N expansion, and has been used to compute the topological expansions of certain matrix integrals.We have recently extended the loop equation formalism to the circular ensembles, where the objects computed are generalisations of matrix integrals over U(N) with Haar measure. The project will involve using some novel tools developed within our group to evaluate these and study their properties.