Construction and Application of Statisitical Mechanics for Random Curves and Patterns

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(1) We defined determinantal processes on spatio-temporal planes as generalizations of determinantal point processes.Using systems of multiple-orthogonal functions, we proved that the noncolliding Brownian motion and the noncolliding Bessel processes are determinantal processes for arbitrary initial configurations with finite numbers of particles, in which all spatio-temporal correlation functions of these processes are determined. We constructed nonequilibrium systems with an infinite number of particles and studied relaxation phenomena to equilibrium states.(2) We introduced a new notion, a complex Brownian motion representation, for the noncolliding Brownian motion. If the system has this representation, we can prove that it is determinantal and all spatio-temporal correlation functions are obtained.(3) We studied O'Connell process, which is related with the quantum Toda lattice. We formulated this proces as a generalization of the noncolliding Brownian motion.