1305.4412 (Makoto Katori)

Determinantal Martingales and Noncolliding Diffusion Processes    [PDF]

Makoto Katori

Noncolliding diffusion processes are dynamical extensions of random matrix models such that any spatio-temporal correlation function is expressed by a determinant and such processes are said to be determinantal. The time-dependent correlation kernel is a functional of initial configuration of the stochastic process and in order to determine it, we have had to deal with multiple orthogonal functions, which should be prepared depending on the initial configuration. From the view point of probability theory, the noncolliding diffusion processes are interesting, since they are realized as the harmonic transforms of absorbing particle systems in the Weyl chambers. Determinantal structure of correlations has not been clear, however, from this view-point. In the present paper, we show direct connections between harmonic transforms and determinantal correlation functions. In this new approach, spatio-temporal correlation kernels can be determined without constructing any system of orthogonal functions. Key quantities are local martingales, which are stochastic processes preserving their expectations in time. By introducing integral transforms, we prove that the harmonic functions of diffusions, which are used for the harmonic transforms, give determinants of matrices whose elements are all martingales. We call them determinantal martingales. We demonstrate how to calculate them depending on processes and initial configurations and show a variety of spatio-temporal correlation kernels are readily derived from them. In special cases, the present martingales are expressed by using complex diffusion processes.

View original: http://arxiv.org/abs/1305.4412