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Distributional limits for critical random graphs
 In preparation
, 2009
"... We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metri ..."
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Cited by 12 (5 self)
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We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean. Keywords: Random graphs, GromovHausdorff distance, scaling limits, continuum random tree, diameter. 2000 Mathematics subject classification: 05C80, 60C05.
9. Bibliography............................................................................111. Team Research Scientist
"... c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1 ..."
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c t i v it y e p o r t 2009 Table of contents 1. Team.................................................................................... 1
ANATOMY OF THE GIANT COMPONENT: THE STRICTLY SUPERCRITICAL REGIME
"... Abstract. In a recent work of the authors and Kim, we derived a complete description of the largest component of the ErdősRényi random graph G(n, p) as it emerges from the critical window, i.e. for p = (1+ε)/n where ε 3 n → ∞ and ε = o(1), in terms of a tractable contiguous model. Here we provide ..."
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Abstract. In a recent work of the authors and Kim, we derived a complete description of the largest component of the ErdősRényi random graph G(n, p) as it emerges from the critical window, i.e. for p = (1+ε)/n where ε 3 n → ∞ and ε = o(1), in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e. the largest component of G(n, p) for p = λ/n where λ> 1 is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2core; Attach i.i.d. Poisson GaltonWatson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim’s Poissoncloning method and the PittelWormald local limit theorems. 1.
The scaling limit of the minimum spanning tree of the complete graph
"... The minimum spanning tree. Consider the complete graph, Kn, on vertices labelled by {1, 2,..., n}. Put independent random weights on the edges which are uniformly distributed on [0, 1] and find the spanning tree Mn of smallest total weight; this is the socalled minimum spanning tree (MST). Now thin ..."
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The minimum spanning tree. Consider the complete graph, Kn, on vertices labelled by {1, 2,..., n}. Put independent random weights on the edges which are uniformly distributed on [0, 1] and find the spanning tree Mn of smallest total weight; this is the socalled minimum spanning tree (MST). Now think of Mn as a metric space by taking the metric to be the graph distance divided by n1/3. We also endow Mn with a probability measure by placing mass 1/n on each vertex. Our main result is the following theorem. Theorem 1. There exists a random compact measured metric space M such that, as n → ∞,
Cycle structure of percolation on highdimensional tori
, 2012
"... In the past years, many properties of the critical behavior of the largest connected components on the highdimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or ErdősRényi rando ..."
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In the past years, many properties of the critical behavior of the largest connected components on the highdimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or ErdősRényi random graph, raising the question whether the scaling limits or the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for highdimensional percolation on the torus {−⌊r/2⌋,..., ⌈r/2 ⌉ − 1} d. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width r/4 centered around each of their vertices, have length of order r d/3, as on the critical ErdősRényi random graph. On the ErdősRényi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by AddarioBerry, Broutin and Goldschmidt (2010). Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on Z d with constraints on their lengths. We believe these estimates are interesting in their own right.