Sharpness of the phase transition for parking on random trees

Contat, Alice

doi:10.1002/rsa.21061

Cited by 7 publications

(9 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that by Proposition 2, the parking process is subcritical if and only if there exists a positive solution to (8). When ( ) holds, since the function x → G(0)xF 0 (x) is strictly increasing, Equation ( 8) has a solution if and only if G(0)x c F 0 (x c ) 2 1 where x c is the radius of convergence of F 0 found in the previous paragraph.…”

Section: Theorem 1: Location Of the Thresholdmentioning

confidence: 87%

“…Since p • = 0, the above calculations show that p • is indeed a solution to the equation (8). Conversely, suppose that there is a positive solution x • to (8). As a special case of equation ( 10) below for F(x, y), we know that the series f(y…”

Section: Decompositionmentioning

confidence: 92%

“…Let µ be a subcritical law for the parking on B. Since p • = 0, the above calculations show that p • is indeed a solution to the equation (8). Conversely, suppose that there is a positive solution x • to (8).…”

Section: Decompositionmentioning

confidence: 95%

“…This triggered an intense activity on the model of parking on a random critical Galton-Watson tree. 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].…”

Section: Introductionmentioning

confidence: 99%

“…The same argument actually even proves that the radius of convergence of X (which is stochastically larger than A) must stay above 2 in the subcritical regime. Notice that deciding whether µ is subcritical for parking depends in a subtle way on the distribution as opposed to the case of critical Galton-Watson trees [14,12,8] where its depends only on the first two moments.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Parking on the infinite binary tree

Aldous¹,

Contat²,

Curien³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Theorem 1: Location Of the Thresholdmentioning

confidence: 87%

Section: Decompositionmentioning

confidence: 92%

Section: Decompositionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Parking on the infinite binary tree

Aldous¹,

Contat²,

Curien³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Parking on the infinite binary tree

Aldous

Contat

Curien

et al. 2023

Probab. Theory Relat. Fields

Self Cite

View full text Add to dashboard Cite

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".

show abstract

Surprising identities for the greedy independent set on Cayley trees

Contat

2022

J. Appl. Probab.

View full text Add to dashboard Cite

We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size n and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of $G_n$ . We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees that is of independent interest.

show abstract

Sharpness of the phase transition for parking on random trees

Cited by 7 publications

References 28 publications

Parking on the infinite binary tree

Parking on the infinite binary tree

Parking on the infinite binary tree

Surprising identities for the greedy independent set on Cayley trees

Contact Info

Product

Resources

About