Some exact correlations in the Dyson Brownian motion model for transitions to the CUE

The Dyson Brownian motion model for transitions to the CUE is considered. For initial eigenvalue probability density functions corresponding to the COE and CSE, the density-density correlation function between an eigenvalue at position λ′ initially (τ = 0) and an eigenvalue at position λ for general τ, is calculated. Theoretical predictions for the asymptotic behaviour are verified, and the distribution for the initial rate of change of the eigenvalues is calculated. Also, for initial conditions corresponding to equispaced eigenvalues, the n-point equal parameter distribution is computed. The extension of the calculation of these quantities to a multiparameter version of the theory is given.