Shape Fluctuations and Random Matrices

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.Comment: Revised version. 51 page

Non-intersecting, simple, symmetric random walks and the extended Hahn kernel

Consider $a$ particles performing simple, symmetric, non-intersecting random walks, starting at points $2(j-1)$, $1\le j\le a$ at time 0 and ending at $2(j-1)+c-b$ at time $b+c$. This can also be interpreted as a random rhombus tiling of an $abc$-hexagon, or as a random boxed planar partition confined to a rectangular box with side lengths $a$, $b$ and $c$. The positions of the particles at all times gives a determinantal point process with a correlation kernel given in terms of the associated Hahn polynomials. In a suitable scaling limit we obtain non-intersecting Brownian motions which can be related to Dysons's Hermitian Brownian motion via a suitable transformation.Comment: 13 page