Geodesic
The phase transition for parking on Galton--Watson trees
Discrete analysis (2022)
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We establish a phase transition for the parking process on critical Galton--Watson trees. In this model, a random number of cars with mean $m$ and variance $σ^{2}$ arrive independently on the vertices of a critical Galton--Watson tree with finite variance $Σ^{2}$ conditioned to be large. The cars go down the tree towards the root and try to park on empty vertices as soon as possible. We show a phase transition depending on $$ Θ:= (1-m)^2- Σ^2 (σ^2+m^2-m).$$ Specifically, when $m \leq 1$, if $ Θ>0,$ then all but (possibly) a few cars will manage to park, whereas if $Θ0$, then a positive fraction of the cars will not find a spot and exit the tree through the root. This confirms a conjecture of Goldschmidt and Przykucki.
Publié le : 
@article{DAS_2022_a18,
     author = {Nicolas Curien and Olivier H\'enard},
     title = {The phase transition for parking on {Galton--Watson} trees},
     journal = {Discrete analysis},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DAS_2022_a18/}
}
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AU  - Nicolas Curien
AU  - Olivier Hénard
TI  - The phase transition for parking on Galton--Watson trees
JO  - Discrete analysis
PY  - 2022
UR  - http://geodesic.mathdoc.fr/item/DAS_2022_a18/
LA  - en
ID  - DAS_2022_a18
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%0 Journal Article
%A Nicolas Curien
%A Olivier Hénard
%T The phase transition for parking on Galton--Watson trees
%J Discrete analysis
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Nicolas Curien; Olivier Hénard. The phase transition for parking on Galton--Watson trees. Discrete analysis (2022). http://geodesic.mathdoc.fr/item/DAS_2022_a18/