One of the new developments in theoretical physics of the past few years has been the application of concepts of string theory to condensed matter systems, the AdS/CFT duality being a prime example of it.Matrix models provide a similar opportunity. In fact, they are ubiquitous in virtually every branch of theoretical physics: from nuclear to condense matter physics, from 2-D quantum gravity to number theory, from statistical physics to string theory, and so on. Even if the physical interpretation of the matrices in the various fields can be quite different, they all share the same mathematical formulation, which makes the possibility of cross-fertilization between the different areas very fruitful.The aim of this project is the study of a class of non-standard matrix models within the context of condensed matter physics, by taking advantage of existing results from string theory. These models, known as weakly confined, are characterized by potentials that asymptotically grow like a log-square and thus do not belong to the (polynomial) Wigner-Dyson universality class, but are still exactly solvable through their orthogonal polynomials.One of the main reasons of interest is that they present a spontaneous breaking of rotational invariance and could therefore be an excellent candidate for studying analytically the Anderson transition between a conducting and an insulating phase, a long standing problem with several experimental applications.Interestingly, models with this asymptotic behavior have been recently considered in the field of topological strings and ABJM theory, but these results have not been translated yet to the condensed matter community and can provide new tools to understand the SSB mechanism.The M.I.T. environment is a perfect and unique place to pursue this kind of research, due to the presence of leading experts in the different areas of physics involved, several of them already involved in similar lines of research.