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We study the noncolliding random walk (RW), which is a particle system ofone-dimensional, simple and symmetric RWs starting from distinct even sites andconditioned never to collide with each other. When the number of particles isfinite, $N < \infty$, this discrete process is constructed as an $h$-transformof absorbing RW in the $N$-dimensional Weyl chamber. We consider Fujita'spolynomial martingales of RW with time-dependent coefficients and express themby introducing a complex Markov process. It is a complexification of RW, inwhich independent increments of its imaginary part are in the hyperbolic secantdistribution, and it gives a discrete-time conformal martingale. The$h$-transform is represented by a determinant of the matrix, whose entries areall polynomial martingales. From this determinantal-martingale representation(DMR) of the process, we prove that the noncolliding RW is determinantal forany initial configuration with $N < \infty$, and determine the correlationkernel as a function of initial configuration. We show that noncolliding RWsstarted at infinite-particle configurations having equidistant spacing arewell-defined as determinantal processes and give DMRs for them. Tracing therelaxation phenomena shown by these infinite-particle systems, we obtain afamily of equilibrium processes parameterized by particle density, which aredeterminantal with the discrete analogues of the extended sine-kernel ofDyson's Brownian motion model with $\beta=2$. Following Donsker's invarianceprinciple, convergence of noncolliding RWs to the Dyson model is alsodiscussed.

Determinantal Martingales and Correlations of Noncolliding Random Walks