The Page-Rényi parking process

In the Page parking (or packing) model on a discrete interval (also known as the discrete Rényi packing problem or the unfriendly seating problem), cars of length two successively park uniformly at random on pairs of adjacent places, until only isolated places remain. We use a probabilistic construction of the Page parking to give a short proof of the (known) fact that the proportion of the interval occupied by cars goes to 1 — e^-2, when the length of the interval goes to infinity. We also obtain some new consequences on both finite and infinite parkings.