Parking on the infinite binary tree

Appears in collection : Probability and Geometry in, on and of non-Euclidian spaces / Probabilités et géométrie dans, sur et des espaces non-euclidiens

Consider a rooted tree whose vertices will be interpreted as free parking spots, each spot accommodating at most one car. On top of that tree, we consider a non-negative integer labeling representing the number of cars arriving on each vertex. Each car tries to park on its arrival vertex, and if the spot is occupied, it travels downwards in direction of the root of the tree until it finds an empty vertex to park. If there is no such vertex on the path towards the root, the car exits the tree, contributing to the flux of cars at the root. This models undergoes an interesting phase transition which we will analyze in detail. After an overview of the case where the underlying tree is a critical Bienayme-Galton-Watson tree, we will concentrate on the case where the underlying tree is the in finite binary tree, where the phase transition turns out to be "discontinuous". The talk is based on a joint work with David Aldous, Nicolas Curien and Olivier Hénard.