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 give a probabilistic 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 obtain some new consequences, and also study a version of this process defined on the infinite line. 1 The Page parking 1.1 The model For n ≥ 2, we consider a sequence of parking configurations xt = (xti)1≤i≤n in {0, 1}n, given by the following construction. Initially the parking is empty: x0 = 0n. Given xt one draws uniformly at random (and independently from the past) a number i in {1, 2,..., n − 1} and, if possible, a car of size 2 parks at i: if xti = xti+1 = 0 then x t+1 i = x t+1 i+1 = 1; n − 2 other coordinates remain unchanged. After some random time Tn (which is dominated by a coupon collector process with n coupons, see Section (3.2) below) parking is no more possible in xTn, in the sense that there is no adjacent coordinates (i, i + 1) such that xTni = x Tn i+1 = 0. We set Xn = x Tn and Xn(i) =