Random covering of the circle: the configuration-space of the free deposition process

Abstract

Consider a circle of circumference 1. Throw at random n points, sequentially, on this circle and append clockwise an arc (or rod) of length s to each such point. The resulting random set (the free gas of rods) is a collection of a random number of clusters with random sizes. It models a free deposition process on a 1D substrate. For such processes, we shall consider the occurrence times (number of rods) and probabilities, as n grows, of the following configurations: those avoiding rod overlap (the hard-rod gas), those for which the largest gap is smaller than rod length s (the packing gas), those (parking configurations) for which hard rod and packing constraints are both fulfilled and covering configurations. Special attention is paid to the statistical properties of each such (rare) configuration in the asymptotic density domain when ns = ρ, for some finite density ρ of points. Using results from spacings in the random division of the circle, explicit large deviation rate functions can be computed in each case from state equations. Lastly, a process consisting in selecting at random one of these specific equilibrium configurations (called the observable) can be modelled. When particularized to the parking model, this system produces parking configurations differently from Renyi's random sequential adsorption model.