Fluctuations of the Occupation Density for a Parking Process

Abstract

ABSTRACT: Consider the following simple parking process on n := {−n,..., n}d , d ≥ 1: at each step, a site i is chosen at random in n and if i and all its nearest neighbor sites are empty, i is occupied. Once occupied, a site remains so forever. The process continues until all sites in n are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of n is called the jamming limit and is denoted by Xn . Ritchie (J Stat Phys 122:381–398, 2006) constructed a stationary random field on Zd obtained as a (thermodynamic) limit of the Xn ’s as n tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box n for the random field X. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case d = 1, in which we also obtain new asymptotic properties for the sequence Xn , n ≥ 1.