Last Car Decomposition of Planar Maps

Contat, Alice

doi:10.48550/arxiv.2205.10285

Cited by 2 publications

(3 citation statements)

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“…Recall that by Proposition 2, the parking process is subcritical if and only if there exists a positive solution to (9). When ( ) holds, since the function x → G(0)xF 0 (x) is strictly increasing, Eq.…”

Section: Remarkmentioning

confidence: 97%

“…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 [6,9,10,15,17] 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.…”

Section: Introductionmentioning

confidence: 99%

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Parking on the infinite binary tree

Aldous

Contat

Curien

et al. 2023

Probab. Theory Relat. Fields

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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 (Bahl et al. in Parking on supercritical Galton-Watson trees, arXiv:1912.13062, 2019 Goldschmidt and Przykucki in Comb Probab Comput 28:23-45, 2019) 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".

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Section: Remarkmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Parking on the infinite binary tree

Aldous

Contat

Curien

et al. 2023

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

Parking on the infinite binary tree

Aldous¹,

Contat²,

Curien³

et al. 2022

Preprint

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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.

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Last Car Decomposition of Planar Maps

Cited by 2 publications

References 0 publications

Parking on the infinite binary tree

Parking on the infinite binary tree

Parking on the infinite binary tree

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