Critical Multi-type Galton–Watson Trees Conditioned to be Large

Abraham, Romain; Delmas, Jean‐François; Guo, Haitao

doi:10.1007/s10959-016-0739-8

Cited by 17 publications

(49 citation statements)

References 18 publications

(47 reference statements)

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“…The difference is that they are focused on the aperiodic case, and that they condition on the vector of population sizes of each types, and not a linear function of it. It is however shown in [3] that, when we condition on only one type, our result can be deduced from theirs.…”

Section: Introductionmentioning

confidence: 74%

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Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

Stephenson

2016

J Theor Probab

100

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We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analogous to Kesten's infinite monotype Galton-Watson tree. This is proven when we condition on the number of vertices of one fixed type, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.

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Section: Introductionmentioning

confidence: 74%

“…In [28] is studied the limit of the multi-type Galton-Watson process associated to the tree, while the authors of [3] are also interested in the local limit of the tree. The difference is that they are focused on the aperiodic case, and that they condition on the vector of population sizes of each types, and not a linear function of it.…”

Section: Introductionmentioning

confidence: 99%

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

Stephenson

2016

J Theor Probab

100

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“…Limits of Galton-Watson trees having a large number of protected nodes were established by Abraham, Bouaziz, and Delmas [1]. The asymptotic shape of conditioned multi-type Galton-Watson trees was studied by Stephenson [18], Abraham, Delmas, and Guo [4], and Pénisson [17].…”

Section: Introductionmentioning

confidence: 99%

Local limits of large Galton–Watson trees rerooted at a random vertex

Stufler¹

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton-Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.

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“…Let Y (α) = (Y s , s ≥ 0) be a strictly stable spectrally positive Lévy process with index α ∈ (1,2] with Laplace exponent…”

Section: Resultsmentioning

confidence: 99%

“…We consider a particular family of multitype GW trees known as alternating two-type GW trees, in which vertices of type 1 only give birth to vertices of type 2 and vice versa. More precisely, given two probability measures µ (1) 2 and µ…”

Section: Alternating Two-type Gw Treementioning

confidence: 99%

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

Berzunza¹

2018

ALEA

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Abstract. In this work, we study asymptotics of multitype Galton-Watson forests with finitely many types. We consider critical and irreducible offspring distributions such that they belong to the domain of attraction of a stable law, where the stability indices may differ. We show that after a proper rescaling, their corresponding height process converges to the continuous-time height process associated with a strictly stable spectrally positive Lévy process. This gives an analog of a result obtained by Miermont (2008) in the case of multitype Galton-Watson forests with finite covariance matrices of the offspring distribution. Our approach relies on a remarkable decomposition for multitype trees into monotype trees introduced in Miermont (2008).

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Critical Multi-type Galton–Watson Trees Conditioned to be Large

Cited by 17 publications

References 18 publications

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

Local Convergence of Large Critical Multi-type Galton–Watson Trees and Applications to Random Maps

Local limits of large Galton–Watson trees rerooted at a random vertex

On scaling limits of multitype Galton-Watson trees with possibly infinite variance

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