A one-dimensional gravitationally interacting gas and the convex minorant of Brownian motion

The surprising connection between a one-dimensional gravitationally interacting gas of sticky particles and the convex minorant process generated by Brownian motion on $[0,1]$ is studied. A study is made of the dynamics of this 1-D gas system by identifying three distinct clustering regimes and the time scales at which they occur. At the critical moment of time the mass distribution of the gas can be computed in terms of functionals of the convex minorant process.