Scaling Limits of Random Graphs from Subcritical Classes

Παναγιώτου, Κωνσταντίνος; Stufler, Benedikt; Weller, Kerstin

doi:10.48550/arxiv.1411.1865

Cited by 4 publications

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“…We naturally distinguish the point o e = p e (0), where p e is the canonical projection, and will usually write T e instead of (T e , d e , o e ). This random metric space (or more precisely its isometry class) appears as the universal scaling limit of many tree-like random objects that naturally appear in combinatorics and probability, see for instance [20] for a survey, and [11,12,19,24,25,26,28,30] for some recent developments on the topic. Here we show that the CRT also appears naturally in this more geometric context.…”

Section: Resultsmentioning

confidence: 99%

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Chen¹,

Miermont

2017

Electron. J. Probab.

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We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of • A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion,• A property of invariance under re-rooting,• The hyperbolicity of the ambient space in the sense of Gromov.A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

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Section: Resultsmentioning

confidence: 99%

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Chen¹,

Miermont

2017

Electron. J. Probab.

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“…More recently, the precise asymptotic estimate has been deduce to be of order Θ( √ n) [36]. Furthermore, the normalized metric space (V (G), d G / √ n) (where d G (u, v) denotes the number of edges in a shortest path that contains u and v in G) is shown to converge with respect to the Gromov-Hausdorff metric to the so-called Brownian Continuum Random Tree multiplied by an scaling factor that depends only the class under study (see [36] for details, and also [40] for extensions to the unlabelled setting). Let us also mention that even more recently, the Schramm-Benjamini convergence had been addressed as well in [20,40] for these graph families.…”

Section: Introductionmentioning

confidence: 99%

Subgraph statistics in subcritical graph classes

Drmota

Ramos

Rué

2017

Random Struct Algorithms

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Let H be a fixed graph and math formula a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in math formula of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series-parallel graphs.Postprint (author's final draft

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“…Suppose that our block-stable class is the class of all series-parallel graphs or another 'subcritical' graph class, or it is the class of planar graphs, or another such class where we know the corresponding generating functions suitably well. In such cases, we may be able to deduce precise asymptotic results, for example about vertex degrees or the numbers and sizes of blocks, by using analytic techniques or by analysing Boltzmann samplers: see for example [2], [6], [7], [8], [10], [11], [12], [13], [14], [22], [23], and for an authoritative recent overview of related work on random planar graphs and beyond see the article [21] by Marc Noy.…”

Section: Introductionmentioning

confidence: 99%