## Critical random graphs: limiting constructions and distributional properties (2010)

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Citations: | 7 - 5 self |

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@MISC{Addario-berry10criticalrandom,

author = {L. Addario-berry and N. Broutin and C. Goldschmidt},

title = {Critical random graphs: limiting constructions and distributional properties },

year = {2010}

}

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### Citations

2044 |
An introduction to probability theory and its applications, volume I
- Feller
- 1968
(Show Context)
Citation Context ... . , Nm(n))n≥0 is performing Pólya’s urn scheme with m colors where the ball picked is replaced along with two extra balls of the same color. It is standard (see, for example, Section VII.4 of Feller =-=[20]-=- or Chapter V, Section 9.1 of Athreya and Ney [8]) that the proportions of balls of each color converge almost surely; indeed, 1 m + 2n (N1(n), N2(n), . . . , Nm(n)) → (N1, N2, . . . , Nm) almost sure... |

1842 | On the evolution of random graphs
- Erdős, Rényi
- 1960
(Show Context)
Citation Context ...troduction The Erdős–Rényi random graph G(n, p) is the graph on vertex set {1, 2, . . . , n} in which each of the ( ) n 2 possible edges is present independently of the others with probability p (see =-=[12, 18, 24]-=-). In a previous paper [1], we considered the rescaled global structure of G(n, p) for p in the critical window – that is, where p = 1/n + λn−4/3 for some λ ∈ R – when individual components are viewed... |

725 | Random Graphs
- Janson, Luczak, et al.
- 2000
(Show Context)
Citation Context ...troduction The Erdős–Rényi random graph G(n, p) is the graph on vertex set {1, 2, . . . , n} in which each of the ( ) n 2 possible edges is present independently of the others with probability p (see =-=[12, 18, 24]-=-). In a previous paper [1], we considered the rescaled global structure of G(n, p) for p in the critical window – that is, where p = 1/n + λn−4/3 for some λ ∈ R – when individual components are viewed... |

453 | Branching processes
- Athreya, Ney
- 1972
(Show Context)
Citation Context ...th m colors where the ball picked is replaced along with two extra balls of the same color. It is standard (see, for example, Section VII.4 of Feller [20] or Chapter V, Section 9.1 of Athreya and Ney =-=[8]-=-) that the proportions of balls of each color converge almost surely; indeed, 1 m + 2n (N1(n), N2(n), . . . , Nm(n)) → (N1, N2, . . . , Nm) almost surely, where (N1, N2, . . . , Nm) ∼ Dirichlet( 1 2 ,... |

154 |
The continuum random tree
- Aldous
(Show Context)
Citation Context ...re-core with the other parts of the tilted tree, and then make the right vertex-identifications. As mentioned in the introduction, we use Aldous’ notion of random finitedimensional distributions (see =-=[4]-=-). By Theorem 3 of [4], the distribution of a continuum random tree coded by an excursion is determined by the distributions of the sequence of finite subtrees obtained by successively sampling indepe... |

124 | Exchangeability and related topics. École d’été de probabilités de Saint-Flour - Aldous - 1985 |

84 | Brownian excursions, critical random graphs and the multiplicative coalescent
- Aldous
- 1997
(Show Context)
Citation Context ...explain this procedure we must first explain the scaling property of the components Ck mentioned above. First, consider the excursions above 0 of the process Bλ . An excursion theory calculation (see =-=[1, 6]-=-) shows that, conditional on their lengths, the distributions of these excursions do not depend on their starting points. Write ˜e (σ) for such an excursion conditioned to have length σ; in the case σ... |

77 |
The continuum random tree. II. An overview
- Aldous
- 1990
(Show Context)
Citation Context ...ss a rather different perspective on real trees with vertex identifications. Suppose first that T is a Brownian CRT. Then the path from the root to a uniformly-chosen leaf has a Rayleigh distribution =-=[3]-=- (see Section 2.2 for a definition of the Rayleigh distribution). This also the distribution of the local time at 0 for a standard Brownian bridge. There is a beautiful correspondence between reflecti... |

57 |
The birth of the giant component,” Random Structures and Algorithms 4
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context ...hat mn −2/3 → 1 and pn → 1. Consider a probability space in which m −1/2 G p m → g(2˜e, P) almost surely as n → ∞; such a space exists by Theorem 1 and Skorohod’s representation theorem. Theorem 7 of =-=[23]-=- implies that for any 3-regular kernel K with fixed surplus k ≥ 2 and with t loops, P (K(G p m) = K|G p m has surplus k) ∝ (1 + o(1)) ( ∏ t 2 e∈E(K) mult(e)! ) −1 , 20as m → ∞. Furthermore, by [27], ... |

51 |
Brownian bridge asymptotics for random mappings
- Aldous, Pitman
- 1994
(Show Context)
Citation Context ...t 0 for a standard Brownian bridge. There is a beautiful correspondence between reflecting Brownian bridge and Brownian excursion given by Bertoin and Pitman [10] (also discussed in Aldous and Pitman =-=[7]-=-), which explains the connection. Let B be a standard reflecting Brownian bridge. Let L be the local time at 0 of B, defined by Let U = sup{t ≤ 1 : Lt = 1 2 L1} and let Kt = Lt = lim ɛ→0 1 2ɛ ∫ t 1 {B... |

47 | Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields
- Evans, Pitman, et al.
(Show Context)
Citation Context ...−x2 /2 . This identifies it as the size-biased Rayleigh distribution, written R ∗ . Finally, UR ∗ has the same distribution as the modulus of a Normal(0, 1) random variable (see p.121 of Evans et al. =-=[19]-=-). Remark. Lemmas 12 and 13 in fact tell us more than we need: they also specify the distribution of the line-segment which originally attached the root to the core. In the case k ≥ 2, this turns out ... |

45 |
Continuous Univariate Distributions, Volume 1
- Johnson, Kotz
- 1970
(Show Context)
Citation Context ...set (X1, X2, . . . , Xn) = j=1 1 ∑n j=1 Γj (Γ1, Γ2, . . . , Γn), then (X1, X2, . . . , Xn) ∼ Dirichlet(α1, α2, . . . , αn), independently of ∑ n j=1 Γj ∼ Gamma( ∑ n j=1 αj, θ) (for a proof see, e.g., =-=[25]-=-). A random variable has Rayleigh distribution if it has density se−s2 /2 on [0, ∞). Note that this is the distribution of the square root of an Exp(1/2) random variable. The significance of the Rayle... |

40 |
Funzione caratteristica di un fenomeno aleatorio. Atti della R. Academia Nazionale dei Lincei, Serie 6. Memorie, Classe di Scienze Fisiche, Mathematice e Naturale, 4:251–299
- Finetti
- 1931
(Show Context)
Citation Context ...proof. We can summarize/rephrase the results of Lemmas 18 and 19 as the following counterpart of the classical limit result for urn models [11, 21] (which is usually proved using de Finetti’s theorem =-=[14]-=-). We do not know of a pre-existing reference for this result in the literature. Theorem 21. Consider the balls-in-urns model described at the beginning of the section, with quantities L1(n), L2(n), .... |

36 |
Path transformations connecting Brownian bridge, excursion and meander
- Bertoin, Pitman
- 1994
(Show Context)
Citation Context ... also the distribution of the local time at 0 for a standard Brownian bridge. There is a beautiful correspondence between reflecting Brownian bridge and Brownian excursion given by Bertoin and Pitman =-=[10]-=- (also discussed in Aldous and Pitman [7]), which explains the connection. Let B be a standard reflecting Brownian bridge. Let L be the local time at 0 of B, defined by Let U = sup{t ≤ 1 : Lt = 1 2 L1... |

34 |
Combinatorial stochastic processes, volume 1875 of Lecture Notes in Mathematics
- Pitman
- 2006
(Show Context)
Citation Context ...n) = 1 + 2 k=1 1 {I(k)=j}, the number of branches corresponding to the jth core edge at step n, for n ≥ 0, 1 ≤ j ≤ m. The following lemma is in the same spirit as Exercises 7.4.11 to 7.4.13 of Pitman =-=[30]-=-. Lemma 16. Conditional on N1(n), . . . , Nm(n), (P1(n), P2(n), . . . , Pm(n)) ∼ Dirichlet(N1(n), N2(n), . . . , Nm(n)). (17) Furthermore, the process (N1(n), . . . , Nm(n))n≥0 evolves as the number o... |

31 |
Certain generalizations in the analysis of variance
- Wilks
- 1932
(Show Context)
Citation Context ..., if A ∼ Gamma(t, 1 2 variables, then from the gamma duplication formula ) and B ∼ Gamma(t + 1 2 , 1 2 ) random variables. ) are independent random AB d = C 2 , (4) where C ∼ Gamma(2t, 1) (see, e.g., =-=[22, 32]-=-). So, we can take Rj = √ Ej, 1 ≤ j ≤ n, where ) and take E1, E2, . . . , En are independent and identically distributed Exp( 1 2 (X1, X2, . . . , Xn) = 1 ∑n j=1 Gj (G1, G2, . . . , Gn), where G1, G2,... |

28 |
Über die Statistik verketteter Vorgänge
- EGGENBERGER, PÓLYA
- 1923
(Show Context)
Citation Context ...of balls at step n of Pólya’s urn model started with with one ball of each color, and evolving in such a way that every ball picked is returned to the urn along with two extra balls of the same color =-=[17]-=-. Then N1(0) = N2(0) = · · · = Nm(0) = 1, and the vector ( ) N1(n) Nm(n) , . . . , m + 2n m + 2n converges almost surely to a limit which has distribution Dirichlet( 1 2 1 , . . . , 2 ) (again, see Se... |

28 | Random trees and applications
- Gall
- 2005
(Show Context)
Citation Context ...oned to have total size σ is essentially that mentioned above, based on vertex identifications within a random real tree. The following presentation owes much to the excellent survey paper of Le Gall =-=[26]-=-. A real tree is a compact metric space (T , d) such that for all x, y ∈ T , • there exists a unique geodesic from x to y i.e. there exists a unique isometry fx,y : [0, d(x, y)] → T such that fx,y(0) ... |

23 |
Un procédé itératif de dénombrement d’arbres binaires et son application à leur génération aléatoire
- Rémy
- 1985
(Show Context)
Citation Context ...s corresponding to balls of colour i is uniform over Catalan trees with 1 + Hi(n) leaves (in fact, if we ignore the edge lengths, this is precisely Rémy’s algorithm to generate a uniform Catalan tree =-=[31]-=-). Furthermore, by Lemma 20, (19) is exactly the right distribution for the relative lengths of edges in the tree AHi(n) created by running the stick-breaking construction of the Brownian CRT for Hi(n... |

21 |
Random Graphs. Cambridge Studies in Advanced Mathematics
- Bollobás
- 2001
(Show Context)
Citation Context ...troduction The Erdős–Rényi random graph G(n, p) is the graph on vertex set {1, 2, . . . , n} in which each of the ( ) n 2 possible edges is present independently of the others with probability p (see =-=[12, 18, 24]-=-). In a previous paper [1], we considered the rescaled global structure of G(n, p) for p in the critical window – that is, where p = 1/n + λn−4/3 for some λ ∈ R – when individual components are viewed... |

16 |
A stochastic approach to the gamma function
- Gordon
- 1994
(Show Context)
Citation Context ..., if A ∼ Gamma(t, 1 2 variables, then from the gamma duplication formula ) and B ∼ Gamma(t + 1 2 , 1 2 ) random variables. ) are independent random AB d = C 2 , (4) where C ∼ Gamma(2t, 1) (see, e.g., =-=[22, 32]-=-). So, we can take Rj = √ Ej, 1 ≤ j ≤ n, where ) and take E1, E2, . . . , En are independent and identically distributed Exp( 1 2 (X1, X2, . . . , Xn) = 1 ∑n j=1 Gj (G1, G2, . . . , Gn), where G1, G2,... |

15 | Recursive self-similarity for random trees, random triangulations and Brownian excursion
- Aldous
- 1994
(Show Context)
Citation Context ...uniformly-chosen vertex in the same tree and with the root of the other, to make a lollipop shape. Finally, we note here an intriguing result which is a corollary of Theorem 6 and Theorem 2 of Aldous =-=[5]-=-. Corollary 9. Take a (rooted) Brownian CRT and sample two uniform leaves. This gives three subtrees, each of which is marked by a leaf (or the root) and the branch-point. These doubly-marked subtrees... |

12 | The continuum limit of critical random graphs. Probab. Theory Related Fields 152
- Addario-Berry, Broutin, et al.
- 2012
(Show Context)
Citation Context ...ddario-Berry ∗ N. Broutin † C. Goldschmidt ‡ August 25, 2009 Abstract We consider the Erdős–Rényi random graph G(n, p) inside the critical window, where p = 1/n + λn −4/3 for some λ ∈ R. We proved in =-=[1]-=- that considering the connected components of G(n, p) as a sequence of metric spaces with the graph distance rescaled by n −1/3 and letting n → ∞ yields a non-trivial sequence of limit metric spaces C... |

12 | The structure of random graph near the point of the phase transition - Luczak, Pittel, et al. - 1994 |

12 | Brownian motion, bridge, excursion and meander characterized by sampling at independent uniform times
- Pitman
- 1999
(Show Context)
Citation Context ... EnGn) = √ ∑n j=1 Gj × 1 √∑n j=1 Gj ( √ E1G1, . . . , √ EnGn), where the products × on each side of the equality involve independent random variables. Applying a Gamma cancellation (Lemma 8 of Pitman =-=[29]-=-), we conclude that √ √ √ (R1 X1, R2 X2, . . . , Rn Xn) d = √ Γ × (Y1, Y2, . . . , Yn), where Γ is independent of (Y1, . . . , Yn) and has a Gamma( n+1 2 1 , 2 ) distribution. 112.3 Distributional pr... |

8 |
The Martin boundary of Pólya’s urn scheme, and an application to stochastic population growth
- Blackwell, Kendall
- 1964
(Show Context)
Citation Context ..., it follows that Xi(n) → √ Pi almost surely, completing the proof. We can summarize/rephrase the results of Lemmas 18 and 19 as the following counterpart of the classical limit result for urn models =-=[11, 21]-=- (which is usually proved using de Finetti’s theorem [14]). We do not know of a pre-existing reference for this result in the literature. Theorem 21. Consider the balls-in-urns model described at the ... |

8 | Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs - Peres, Revelle - 2004 |

7 | Dual random fragmentation and coagulation and an application to the genealogy of Yule processes
- Bertoin, Goldschmidt
- 2004
(Show Context)
Citation Context ... cycle and the second to be the length of the segment attaching it to the core. The results stated in the following lemma are straightforward and may be found, for example, in Bertoin and Goldschmidt =-=[9]-=-. Lemma 13. Suppose that (Y1, Y2, . . . , Ym) ∼ Dirichlet(1, 1, . . . , 1). Let Y ∗ be a size-biased pick from this vector, and relabel the other co-ordinates arbitrarily Y ∗ 1 , Y ∗ 2 , . . . , Y ∗ m... |

7 | Anatomy of a young giant component in the random graph, preprint, available at http://arxiv.org/abs/0906.1839 14
- Ding, Kim, et al.
(Show Context)
Citation Context ... a core edge” converges to reflected Brownian motion. In the barely supercritical case (i.e. in G(n, p) when p = (1 + ɛ(n))/n and n 1/3 ɛ(n) → ∞ but ɛ(n) = o(n −1/4 )), Ding, Kim, Lubetzky, and Peres =-=[15]-=- have shown that the “edge trees” of the largest component of G(n, p) may essentially be generated by the following procedure: start from a path of length given by a geometric with parameter ɛ, then a... |

5 |
Algebraic properties of beta and gamma distributions, and applications
- Dufresne
- 1998
(Show Context)
Citation Context ...e say that a random variable has a Beta(a, b) distribution if it has density Γ(a + b) Γ(a)Γ(b) xa−1 (1 − x) b−1 on [0, 1]. We will make considerable use of the so-called beta-gamma algebra (see [13], =-=[16]-=-), which consists of a collection of distributional relationships which may be summarized as Gamma(α, θ) d = Gamma(α + β, θ) × Beta(α, β), where the terms on the right-hand side are independent. We wi... |

4 |
friedman’s urn
- Bernard
- 1965
(Show Context)
Citation Context ..., it follows that Xi(n) → √ Pi almost surely, completing the proof. We can summarize/rephrase the results of Lemmas 18 and 19 as the following counterpart of the classical limit result for urn models =-=[11, 21]-=- (which is usually proved using de Finetti’s theorem [14]). We do not know of a pre-existing reference for this result in the literature. Theorem 21. Consider the balls-in-urns model described at the ... |

4 |
The structure of random graphs at the point of transition
- Luczak, Pittel, et al.
- 1994
(Show Context)
Citation Context ...f [23] implies that for any 3-regular kernel K with fixed surplus k ≥ 2 and with t loops, P (K(G p m) = K|G p m has surplus k) ∝ (1 + o(1)) ( ∏ t 2 e∈E(K) mult(e)! ) −1 , 20as m → ∞. Furthermore, by =-=[27]-=-, Theorem 4, all vertices of degree three in K(Gp m) are separated by distance of order m1/2 , so all such vertices remain distinct in the limit. This proves that the shape of the limiting kernel K(2˜... |

4 | Weak convergence of random p-mappings and the exploration process of inhomogeneous continuum random trees - Aldous, Miermont, et al. |

4 | The loop-erased random walk and the uniform spanning tree on the four-dimensional discrete torus. Preprint, available at http://front.math.ucdavis.edu/ math.PR/0602515 - Schweinsberg - 2006 |

2 | Exercises in probability, volume 13 of Cambridge Series in Statistical and Probabilistic Mathematics - Chaumont, Yor - 2003 |

1 | ISSN 0091-1798. MR1288122 D. Aldous. Brownian excursions, critical random graphs and the multiplicative coalescent - Probab - 1994 |

1 | Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs - Soc - 1994 |