Parking on the infinite binary tree

Abstract

Let (A u : u ∈ B) be i.i.d. non-negative integers that we interpret as car arrivals on the vertices of the full binary tree B. Each car tries to park on its arrival node, but if it is already occupied, it drives towards the root and parks on the first available spot. It is known [14,1] that the parking process on B exhibits a phase transition in the sense that either a finite number of cars do not manage to park in expectation (subcritical regime) or all vertices of the tree contain a car and infinitely many cars do not manage to park (supercritical regime). We characterize those regimes in terms of the law of A in an explicit way. We also study in detail the critical regime as well as the phase transition which turns out to be "discontinuous".Figure 1: Simulation of the parking process on the first 9 levels of the full binary tree (the edges are oriented towards the origin of the tree which is at the center of the figures). The dotted vertices on the left figure initially contain 2 cars whereas the others are void. In the middle and right figures, the highlighted edges have seen a positive flux of car.