  We study systems of interacting Brownian particles in one dimension
constructed as the diffusion scaling limits of Fisher's vicious walk models. We
define two types of nonintersecting Brownian motions, in which we impose no
condition (resp. nonintersecting condition forever) in the future for the
first-type (resp. second-type). It is shown that, when all particles start from
the origin, their positions at time 1 in the first-type (resp. at time 1/2 in
the second-type) process are identically distributed with the eigenvalues of
Gaussian orthogonal (resp.unitary) random matrices. The second-type process is
described by the stochastic differential equations of the Dyson-type Brownian
motions with repulsive two-body forces proportional to the inverse of
distances. The present study demonstrates that the spatio-temporal coarse
graining of random walk models with contact interactions can provide many-body
systems with long-range interactions.
