Correlations for the circular Dyson brownian motion model with Poisson initial conditions

The circular Dyson brownian motion model refers to the stochastic dynamics of the log-gas on a circle. It also specifies the eigenvalues of certain pa rameter-dependent ensembles of unitary random matrices. This model is consi dered with the initial condition that the particles are noninteracting (Poi sson statistics). Jack polynomial theory is used to derive a simple exact e xpression for the density-density correlation with the position of one part icle specified in the initial state, and the position of one particle speci fied at time tau, valid for all beta > 0, The same correlation with two par ticles specified in the initial state is also derived exactly, and some spe cial cases of the theoretical correlations are illustrated by comparison wi th the empirical correlations calculated from the eigenvalues of certain pa rameter-dependent Gaussian random matrices. Application to fluctuation form ulas for time-displaced linear statistics in made. (C) 1998 Elsevier Scienc e B.V.