"Dynamical correlations among vicious random walkers",
( with Nagao, T. and Katori, M.)
Phys. Lett. A 307/1 (2003), 29-35
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Nonintersecting motion of Brownian particles in one dimension is studied.
The system is constructed as the diffusion scaling limit of Fisher's vicious
random walk. N particles start from the origin at time t=0 and then undergo
mutually avoiding Brownian motion until a finite time t=T. In the short time
limit $t \ll T$, the particle distribution is asymptotically described by
Gaussian Unitary Ensemble (GUE) of random matrices. At the end time t = T,
it is identical to that of Gaussian Orthogonal Ensemble (GOE). The Brownian
motion is generally described by the dynamical correlations among particles
at many times $t_1,t_2,..., t_M$ between t=0 and t=T. We show that the most
general dynamical correlations among arbitrary number of particles at arbitrary
number of times are written in the forms of quaternion determinants.
Asymptotic forms of the correlations in the limit $N \to \infty$ are
evaluated and a discontinuous transition of the universality class
from GUE to GOE is observed.