Leslie Molag

Extensions of interacting particle systems to higher dimensions

In the 60s Freeman Dyson introduced the idea to replace the Hamiltonians of certain disordered systems by large random matrices. Based on physical symmetries, he distinguished three categories. These concern the Gaussian unitary/orthogonal/symplectic ensembles. Their eigenvalues are real and are related to the energy levels. In 1965, motivated by mathematical curiosity, Jean Ginibre introduced counterparts to these three models whose eigenvalues live in the complex plane. When one properly rescales the eigenvalues of these random matrix models one finds various limits (scaling limits) that turn out to be universal. Recently, it was realized that non-interacting fermions in higher dimensions are essentially a higher dimensional version of the Gaussian unitary ensemble, with its own set of scaling limits. We end the talk by introducing a counterpart of this model that lives in higher complex dimension, and present some of its scaling limits.