Brownian-motion model of discrete spectra

The transition in spectral fluctuations in a quantum chaotic system as a good symmetry of the system is gradually broken is describable in terms of Dysonʹs Brownian-motion model. Here the magnitude of symmetry breaking plays the role of ‘time’ while the ‘temperature’ determines the universality class of random matrices to which the system goes in ‘equilibrium’. Recently we have shown that the transition itself has a universal classification and is useful in studying exact or weakly broken symmetries in complex many-body systems. In this paper we derive the exact two-level correlation functions for the class of transitions which eventually go to the Gaussian unitary ensemble. It appears that the transition to the Gaussian orthogonal and symplectic ensembles may also be exactly solvable.