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)$(N\times N)$-matrices consisting of a single _sl2${\mathit{sl}}_2$-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)$H_N=\mathit{SL}(N,R)/\mathit{SO}(N)$. 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$H_N$. The coordinates of _HN$H_N$ 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.