• one dimensonal physics
• third order phase transition
• random matrices
• large deviations
• extreme value statistics
• fisica statistica

Random matrix theory (RMT) offers a powerful theoretical framework to model disordered
systems. The eigenvalues of a symmetric random matrix ensemble can be
seen as a long range interacting one dimensional gas. The extreme value statistics
of this kind of gases present a rich behavior: the limiting distribution of the typical
fluctuations might not lie in one of the 3 classical universality classes; and the
large deviations shed light to the so called pushed-to-pulled phase transition, a 3rd
order phase transition in which the order parameter is the volume of the gas. We
present a model of a one dimensional gas interacting via the 1D Coulomb potential
and confined by a potential inspired by the Wishart ensemble of RMT. We are able
to provide analytical results for the limiting distribution and the large deviations of
the closest and the farthest particle from the origin. Furthermore, what makes this
toy model really interesting is that we could find an analytical expression for the
partition functions associated to the cumulative of the closest and the farthest particle
at finite N. That allowed us to compute the full finite N order statistics and the
finite N empirical distributions. We also provide numerical checks for the finite N
calculations via Montecarlo algorithms.