"Noncolliding Brownian Motion and Determinantal Processes",
A system of one-dimensional Brownian motions (BMs)
conditioned never to collide with each other
is realized as (i) Dyson's BM model,
which is a process of eigenvalues of hermitian
matrix-valued diffusion process in the
Gaussian unitary ensemble (GUE),
and as (ii) the $h$-transform of absorbing BM
in a Weyl chamber, where the harmonic function $h$
is the product of differences
of variables (the Vandermonde determinant).
The Karlin-McGregor formula gives determinantal
expression to the transition probability density
of absorbing BM. We show from the
Karlin-McGregor formula, if the initial state
is in the eigenvalue distribution of GUE,
the noncolliding BM is a determinantal process,
in the sense that any multitime correlation function is
given by a determinant specified by a matrix-kernel.
By taking appropriate scaling limits,
spatially homogeneous and inhomogeneous
infinite determinantal processes are derived.
We note that the determinantal processes related with
noncolliding particle systems have a feature in common
such that the matrix-kernels are expressed
using spectral projections of
appropriate effective Hamiltonians.
On the common structure of matrix-kernels,
continuity of processes in time
is proved and general property of the
determinantal processes is discussed.
