Fluctuations of the Occupation Density for a Parking Process

Consider the following simple parking process on \(\Lambda _n:= \{-n, \ldots , n\}^d,d\ge 1\): at each step, a site i is chosen at random in \(\Lambda _n\) 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 \(\Lambda _n\) are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of \(\Lambda _n\) is called the jamming limit and is denoted by \(X_{\Lambda _n}\). Ritchie (J Stat Phys 122:381–398, 2006) constructed a stationary random field on \(\mathbb {Z}^d\) obtained as a (thermodynamic) limit of the \(X_{\Lambda _n}\)’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 \(\Lambda _n\) 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 \(X_{\Lambda _n},n\ge 1\).