Author:
De Ambroggio Umberto,Roberts Matthew I.
Abstract
AbstractWe 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science
Cited by
2 articles.
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