## Determinantal Martingales and Correlations of Noncolliding Random Walks    [PDF]

Makoto Katori
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 forever. When the number of particles is finite, $N < \infty$, this discrete process is constructed as an $h$-transform of an 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 process. 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. Based on reducibility and consistency of the DMRs, we define a family of noncolliding RWs with infinite numbers of particles. Tracing the relaxation phenomena of the 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 $\beta=2$.
View original: http://arxiv.org/abs/1307.1856