An elementary approach to component sizes in critical random graphs

Ambroggio, Umberto De

doi:10.1017/jpr.2022.13

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2022

2024

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Cited by 4 publications

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“…In future work we intend to demonstrate the adaptability of our new approach by applying our methods to other random graph models. As a first step in this direction, for applications of Lemma 1.2 to a random intersection graph, an inhomogeneous random graph, and percolation on a d-regular graph, see [10].…”

Section: Introductionmentioning

confidence: 99%

Unusually large components in near-critical Erdős–Rényi graphs via ballot theorems

Ambroggio

Roberts

2022

Combinator. Probab. Comp.

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We consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form $p=p(n)=1/n+\lambda/n^{4/3}$ and A is large, \begin{equation*}\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}},\end{equation*} where $\mathcal{C}_{\max}$ is the largest connected component of the graph. Our result allows A and $\lambda$ to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.

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Section: Introductionmentioning

confidence: 99%