  Determinantal process is a dynamical extension of a determinantal point
process such that any spatio-temporal correlation function is given by a
determinant specified by a single continuous function called the correlation
kernel. Noncolliding diffusion processes are important examples of
determinantal processes. In the present lecture, we introduce determinantal
martingales and show that if the interacting particle system (IPS) has
determinantal-martingale representation, then it becomes a determinantal
process. From this point of view, the reason why noncolliding diffusion
processes and noncolliding random walk are determinantal is simply explained by
the fact that the harmonic transform with the Vandermonde determinant provides
a proper determinantal martingale. Recently O'Connell introduced an interesting
IPS, which can be regarded as a stochastic version of a quantum Toda lattice.
It is a geometric lifting of the noncolliding Brownian motion and is not
determinantal, but Borodin and Corwin discovered a determinant formula for a
special observable for it. We also discuss this new topic from the present
view-point of determinantal martingale.
