  We study the noncolliding random walk (RW), which is a particle system of
one-dimensional, simple and symmetric RWs starting from distinct even sites and
conditioned never to collide with each other. When the number of particles is
finite, $N < \infty$, this discrete process is constructed as an $h$-transform
of absorbing RW in the $N$-dimensional Weyl chamber. We consider Fujita's
polynomial martingales of RW with time-dependent coefficients and express them
by introducing a complex Markov process. It is a complexification of RW, in
which independent increments of its imaginary part are in the hyperbolic secant
distribution, and it gives a discrete-time conformal martingale. The
$h$-transform is represented by a determinant of the matrix, whose entries are
all polynomial martingales. From this determinantal-martingale representation
(DMR) of the process, we prove that the noncolliding RW is determinantal for
any initial configuration with $N < \infty$, and determine the correlation
kernel as a function of initial configuration. We show that noncolliding RWs
started at infinite-particle configurations having equidistant spacing are
well-defined as determinantal processes and give DMRs for them. Tracing the
relaxation phenomena shown by these infinite-particle systems, we obtain a
family of equilibrium processes parameterized by particle density, which are
determinantal with the discrete analogues of the extended sine-kernel of
Dyson's Brownian motion model with $\beta=2$. Following Donsker's invariance
principle, convergence of noncolliding RWs to the Dyson model is also
discussed.
