Random ballistic growth and diffusion in symmetric spaces

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N-column box is viewed as a time-ordered product of (N×N)-matrices consisting of a single sl2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space HN=SL(N,R)/SO(N). In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in HN. The coordinates of HN are interpreted as the coordinates of particles of the one-dimensional Toda chain. The group-theoretic structure of the system and links to some random matrix models are also discussed.