Stretched exponential decay for subcritical parking times on ℤd. Issue 2 (9th February 2021) - Damron, Michael; Lyu, Hanbaek; Sivakoff, David
Abstract: At each vertex of ℤ d, place a car with probability p or vacant parking spot with probability 1 − p . Cars perform independent random walks and park at vacant spots, rendering them passable. There is a transition at p = 1/2: the origin is a.s. visited by finitely many distinct cars when p < 1/2, and by infinitely many when p ≥ 1/2. Furthermore, a.s. all cars park if p ≤ 1/2 and some never park for p > 1/2. For small p, we prove that the parking time τ of the car initially at the origin satisfies exp − C 1 t d d + 2 ≤ ℙ p ( τ > t ) ≤ exp − c 2 t d d + 2 . For d = 1, these inequalities hold for p < 1/2. In contrast, when p > 1/2 and d = 1, methods of Bramson–Lebowitz imply that the tail of the parking time of the spot of the origin decays like e − c t . Our exponent d /( d + 2) also differs from those previously obtained in the case of moving obstacles.