Limit theory for random sequential packing and deposition (Q1872410)

The authors consider the following basic random packing model. Unit volume open balls \(B_{1,n},B_{2,n},\dots\) arrive sequentially and uniformly at random in the \(d\)-dimensional cube having volume \(n\) and centered at the origin. Let the first ball \(B_{1,n}\) be packed, and recursively for \(i = 2,3,\dots\), let the \(i\)th ball \(B_{i,n}\) be packed iff \(B_{i,n}\) does not overlap any ball in \(B_{1,n},\dots,B_{i-1,n}\) which has already been packed. If not packed, the \(i\)th ball is discarded. Denote by \(N_{n,d}(k)\) the number of balls packed out of the first \(k\) arrivals. The attention is restricted to the fixed input packing number \(N_{n,d}(k)\), where \(k\) is a given positive integer, and to the Poisson input packing number \(N_{n,d}(\text{Po}(\lambda))\), where Po\((\lambda)\) is an independent Poisson random variable with parameter \(\lambda\). The authors prove a law of large numbers and central limit theorem for the packing number \(N_{n,d}\), as \(n\to\infty\), in the thermodynamic limit with the (expected) number of incoming balls proportional to \(n\). The paper also contains analogous limit results for four variants of the basic random packing model: (i) models with unit volume balls replaced by particles of random size/shape/charge, (ii) time dependent models, (iii) cooperative sequential adsorption models, and (iv) ballistic decomposition models. The proofs are based on a general law of large numbers and central limit theorem for functionals of marked binomial and Poisson point processes.