Parking on Cayley trees & Frozen Erdös-Rényi

Abstract

Consider a uniform rooted Cayley tree T n with n vertices and let m cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner & Panholzer [56] established a phase transition for this process when m ≈ n 2 . In this work, we couple this model with a variant of the classical Erdős-Rényi random graph process. This enables us to describe the phase transition for the size of the components of parked cars using a modification of the multiplicative coalescent which we name the frozen multiplicative coalescent. The geometry of critical parked clusters is also studied. Those trees are very different from Bienaymé-Galton-Watson trees and should converge towards the growth-fragmentation trees canonically associated to the 3/2-stable process that already appeared in the study of random planar maps.

Parking on the infinite binary tree

“…Section 4 is devoted to the study of F via a functional equation obtained by splitting a fully parked tree at the root, see (10). But before doing so, let us present the combinatorial decomposition and the characterization of subcriticality in terms of F. It turns out that most equations simplify if one introduces F(x, y) := 1 + F(x, y) and F 0 (x) := F(x, 0) := 1 + F(x, 0).…”

“…To solve this equation, we apply the kernel method of Bousquet-Mélou and Jehanne [4] and look for a (formal) power series Y = Y(x) such that ∂ f P(F(x, Y(x)), F 0 (x), x, Y(x)) = 0 so that combined with (10) we also find automatically ∂ y P(F(x, Y(x)), F 0 (x), x, Y(x)) = 0. This introduction may seem ad-hoc, but it enables us to find a system of three equations on the three unknowns F, F 0 and Y, so that with a little luck we will find an "expression" for those.…”

“…In this section we use the explicit resolution of the functional equation (10) to determine the radius of convergence x c of F 0 and the value of F 0 (x c ). The important fact for our application to parking being that under the condition ( ) on the existence of t c in Theorem 1 we have…”

“…In particular, a phase transition was proved to occur and the threshold was located in an increasing level of generality [14,12,8]. Furthermore a surprising connection with the Erdös-Rényi random graph and the multiplicative coalescent was unraveled in [10].…”

“…Theorem 1 boils down to deciding whether we have a non trivial solution p • = 0 to these equations (otherwise we are in the supercritical regime). The critical regime corresponds to the case where p • is exactly the radius of convergence of F. Thus the main ingredient in the proof is the "computation" of the generating function F. The enumeration of fully parked trees has already been considered in the combinatorics literature [17,6,15,10,9] and it shares many similarities with the enumeration of planar maps. The idea is to enumerate a more complicated structure, namely fully parked trees with a possible flux of cars at the origin.…”