Abstract Consider the following simple parking process on $$\Lambda _n:= \{-n, \ldots , n\}^d,d\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>{</mml:mo> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> : at each step, a site i is chosen at random in $$\Lambda _n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> 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 $$\Lambda _n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of $$\Lambda _n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> is called the jamming limit and is denoted by $$X_{\Lambda _n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msub> </mml:math> . Ritchie (J Stat Phys 122:381–398, 2006) constructed a stationary random field on $$\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> obtained as a (thermodynamic) limit of the $$X_{\Lambda _n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msub> </mml:math> ’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 $$\Lambda _n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> 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 $$d=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , in which we also obtain new asymptotic properties for the sequence $$X_{\Lambda _n},n\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> .