Transitions between universality classes of random matrices

Abstract

Dyson’s Brownian-motion model is recovered as a rigorous description of the dependence of Hamiltonians H=${\mathit{H}}_0$+λV on the control parameter λ when ${\mathit{H}}_0$ and V belong to different universality classes. For instance, time-reversal invariance may hold for ${\mathit{H}}_0$ but not for V. The clue to our result lies in (i) a certain rescaling of the energy, H→ √f(λ) (${\mathit{H}}_0$+λV), and (ii) not individual Hamiltonians but suitable families.