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Limits of multiplicative inhomogeneous random graphs and Lévy trees: Limit theorems [article]

Nicolas Broutin, Thomas Duquesne, Minmin Wang
2020 arXiv   pre-print
We consider a natural model of inhomogeneous random graphs that extends the classical Erd\H os-R\'enyi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous  ...  This embedding transfers asymptotically into an embedding of the limit objects into a forest of L\'evy trees, which allows us to give an explicit construction of the limit objects from the excursions of  ...  the study of scaling limits of critical random graphs.  ... 
arXiv:2002.02769v1 fatcat:wkcua7npnzeirfzkrjvlvqpim4

Limits of multiplicative inhomogeneous random graphs and Lévy trees: The continuum graphs [article]

Nicolas Broutin, Thomas Duquesne, Minmin Wang
2020 arXiv   pre-print
limit of graphs related to the multiplicative coalescent (the Erdős–Rényi random graph, more generally the so-called rank-one inhomogeneous random graphs of various types, and the configuration model)  ...  In a companion paper, we show that the continuous Lévy graphs are indeed the scaling limit of inhomogeneous random graphs.  ...  [17] (packing case) in Euclidian spaces that have been extended in Edgar [12] (see Theorem 4.15 and Proposition 4.24 for the Hausdorff case and see Theorem 5.9 for the packing case).  ... 
arXiv:1804.05871v2 fatcat:7furehdnuzfv3fyd2bz6oyue2e

Central limit theorems for local network statistics [article]

P-A. Maugis
2020 arXiv   pre-print
We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model which generalizes many popular statistical network models.  ...  Subgraph counts - in particular the number of occurrences of small shapes such as triangles - characterize properties of random networks, and as a result have seen wide use as network summary statistics  ...  Acknowledgements I gratefully acknowledge that the authors of [47] kindly shared the dataset used in this work, and provided comments on early versions of this document. Discussions  ... 
arXiv:2006.15738v1 fatcat:bmrlddbvy5hwzhju372go5efue

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs [article]

Shankar Bhamidi, Remco van der Hofstad, Sanchayan Sen
2017 arXiv   pre-print
The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with  ...  One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as  ...  ACKNOWLEDGEMENTS The authors are indebted to Grégory Miermont for many enlightening discussions about inhomogeneous continuum random trees.  ... 
arXiv:1508.04645v3 fatcat:j5jcltastretjhqa5oie7tzs5q

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Nicolas Broutin, Minmin Wang
2017 Bernoulli  
The theorems for the inhomogeneous continuum random tree extend previous results by Bertoin and Miermont about the cut tree of the Brownian continuum random tree. that for the random Cayley trees, L n  ...  We then use these results to study the fragmentation of the inhomogeneous continuum random trees (scaling limits of p-trees) and give distributional correspondences between the initial tree and the tree  ...  Acknowledgements We thank the referees for their careful reading and helpful suggestions.  ... 
doi:10.3150/16-bej813 fatcat:lxux6gulm5codjhwqcbuhmi3bi

Parking on Cayley trees Frozen Erdös-Rényi [article]

Alice Contat, Nicolas Curien
2021 arXiv   pre-print
In this work, we couple this model with a variant of the classical Erdös-Rényi random graph process.  ...  study of random planar maps.  ...  We thank Linxiao Chen, Armand Riera and especially Olivier Hénard for several motivating discussions during the elaboration of this work.  ... 
arXiv:2107.02116v1 fatcat:vvk47gyiszbz7olf72yagq5d7u

Cutting down p-trees and inhomogeneous continuum random trees [article]

Nicolas Broutin, Minmin Wang
2014 arXiv   pre-print
The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree.  ...  We then use these results to study the fragmentation of the ICRTs (scaling limits of p-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation.  ...  Inhomogeneous continuum random trees The inhomogeneous continuum random tree (abbreviated as ICRT in the following) has been introduced in [21] and [8] .  ... 
arXiv:1408.0144v3 fatcat:jofc3e4ozfb6xc4774tkqyxqty

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

Shankar Bhamidi, Remco van der Hofstad, Sanchayan Sen
2017 Probability theory and related fields  
and Limic Electron in J Probab 3(3):59, 1998), yielding a completely new class of limiting random metric spaces.  ...  A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent τ ∈ (3, 4) where edges are rescaled by n −(τ −3)/(τ −1) yielding the  ...  Acknowledgements The authors are indebted to Grégory Miermont for many enlightening discussions about inhomogeneous continuum random trees.  ... 
doi:10.1007/s00440-017-0760-6 fatcat:mqtt4v3mxbft5nk7sxaxu7ka2e

Decorated stable trees [article]

Delphin Sénizergues, Sigurdur Örn Stefánsson, Benedikt Stufler
2022 arXiv   pre-print
We mention a few examples where those objects appear in the context of random trees and planar maps, and we expect them to naturally arise in many more cases.  ...  We prove an invariance principle which states that under some conditions, analogous discrete objects, random decorated discrete trees, converge in the scaling limit to decorated α-stable trees.  ...  Its Gromov-Hausdorff limit then generically turns out to be a multiple of the Brownian tree itself (see [36, Theorem 6 .60]). Outline The paper is organized as follows.  ... 
arXiv:2205.02968v1 fatcat:a5kdtq2vdzcrld6yw26g6nhhzu

Invariance principles for random walks in random environment on trees [article]

George Andriopoulos
2020 arXiv   pre-print
To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees.  ...  In arXiv:1609.05666v1 [math.PR] a functional limit theorem was proved.  ...  Acknowledgements I would like to thank my supervisor Dr David Croydon for suggesting the problem, his support and many useful discussions.  ... 
arXiv:1812.10197v2 fatcat:uqbf36b4wfgwjet4abfseyxmdy

Invariance principles for pruning processes of Galton-Watson trees [article]

Hui He, Matthias Winkel
2014 arXiv   pre-print
Pruning processes (F(θ),θ≥ 0) have been studied separately for Galton-Watson trees and for Lévy trees/forests. We establish here a limit theory that strongly connects the two studies.  ...  topology on cadlag functions with values in the space of (isometry classes of) locally compact real trees to limiting pruning processes.  ...  Acknowledgements This work was started during a research visit of the second author to Beijing Normal University.  ... 
arXiv:1409.1014v1 fatcat:arkk4vz66rgrjhlstikiuuccmy

Random ultrametric trees and applications

Amaury Lambert, Jean-François Coeurjolly, Adeline Leclercq-Samson
2017 ESAIM Proceedings and Surveys  
We display a variety of examples of random combs and explain how they can be used in applications.  ...  In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree. Résumé.  ...  It is an open question to investigate whether the compactification mentioned earlier in this subsection of the tree into a CPP can shed extra light on these results.  ... 
doi:10.1051/proc/201760070 fatcat:owhi7eowcvh2rji2zsv5v3cxv4

Scaling limits for some random trees constructed inhomogeneously [article]

Nathan Ross, Yuting Wen
2016 arXiv   pre-print
We define some new sequences of recursively constructed random combinatorial trees, and show that, after properly rescaling graph distance and equipping the trees with the uniform measure on vertices,  ...  The limiting real trees are constructed via line-breaking the real half-line with a Poisson process having rate (ℓ+1)t^ℓ dt, for each positive integer ℓ, and the growth of the combinatorial trees may be  ...  Acknowledgement We thank Adrian Röllin for suggesting inserting random vertices into the real trees.  ... 
arXiv:1611.01306v1 fatcat:b3xifets4vdphbefvqj4plq5hq

The coalescent point process of branching trees

Amaury Lambert, Lea Popovic
2013 The Annals of Applied Probability  
This process of point measures is also closely related to an inhomogeneous spine decomposition of the lineage of the first surviving particle in generation h in a planar BGW tree conditioned to survive  ...  The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process (A_i; i> 1), where A_i is the coalescence time between individuals i and i+1.  ...  Lambert wishes to thank Julien Berestycki and Olivier Hénard for some interesting discussions on the topic of this paper. A. Lambert and L.  ... 
doi:10.1214/11-aap820 fatcat:zcodm4srkzbfxatsutwqqdtfje

Inverting the cut-tree transform [article]

Louigi Addario-Berry, Daphné Dieuleveut, Christina Goldschmidt
2016 arXiv   pre-print
The cut-tree was generalised by Dieuleveut to a fragmentation of the α-stable trees, α∈ (1, 2), and by Broutin and Wang to the inhomogeneous continuum random trees of Aldous and Pitman.  ...  The genealogy of such a fragmentation is encoded by the so-called cut-tree, which was introduced by Bertoin and Miermont for a fragmentation of the Brownian continuum random tree.  ...  We are very grateful to Edward Crane for sharing the results of [23] with us.  ... 
arXiv:1606.04825v2 fatcat:lf5xbxukhbcvvekmf6gj52t62i
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