Random ballistic growth and diffusion in symmetric spaces

TL;DR

This paper explores the connection between ballistic deposition growth processes and diffusion in symmetric spaces, revealing links to random matrix models and integrable systems like the Toda chain.

Contribution

It establishes a novel relation between surface growth in ballistic deposition and diffusion in symmetric spaces, integrating group theory and random matrix theory.

Findings

Distribution of maximal heap height relates to diffusion in H_N.

Growth process modeled as product of random matrices with SL(N,R)/SO(N) structure.

Connections made to Toda chain particles and random matrix models.

Abstract

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N-column box is viewed a time-ordered product of N\times N-matrices consisting of a single sl_2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space H_N=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 H_N. The coordinates of 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.