1206.2185 (Makoto Katori)

System of Complex Brownian Motions Associated with the O'Connell Process    [PDF]

Makoto Katori

The O'Connell process is a softened version (a geometric lifting with a parameter $a>0$) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length $a$. This process is not determinantal in general. Under the special initial condition, however, Borodin and Corwin gave a Fredholm determinantal expression to the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and give a complex Brownian motion (CBM) representation to the observable. The complex function parameterized by the drift vector, which gives the determinantal expression to the weight of CMB paths, is not entire, but it becomes an entire function providing conformal martingales in the tropicalization $a \to 0$.

View original: http://arxiv.org/abs/1206.2185