Random sequential adsorption of discs on constant-curvature surfaces: plane, sphere, hyperboloid, and projective plane

Abstract

We present an algorithm to simulate random sequential adsorption (random "parking") of discs on constant-curvature surfaces: the plane, sphere, hyperboloid, and projective plane, all embedded in three-dimensional space. We simulate complete parkings by explicitly calculating the boundary of the available area in which discs can park and concentrating new points in this area. This makes our algorithm efficient and also provides a diagnostic to determine when each parking is complete, so there is no need to extrapolate data from incomplete parkings to study questions of physical interest. We use our algorithm to study the number distribution and density of discs parked in each space, where for the plane and hyperboloid we consider two different periodic tilings each. We make several notable observations: (i) On the sphere, there is a critical disc radius such the number of discs parked is always exactly four: the random parking is actually deterministic. We prove this statement rigorously, and also show that random parking on the surface of a $d$-dimensional sphere would have deterministic behaviour at the same critical radius. (ii) The average number of parked discs does not always monotonically increase as the disc radius decreases: on the plane (square with periodic boundary conditions), there is an interval of decreasing radius over which the average *decreases*. We give a heuristic explanation for this counterintuitive finding. (iii) As the disc radius shrinks to zero, the density (average fraction of area covered by parked discs) appears to converge to the same constant for all spaces, though it is always slightly larger for a sphere and slightly smaller for a hyperboloid. Therefore, for parkings on a general curved surface we would expect higher local densities in regions of positive curvature and lower local densities in regions of negative curvature.