Geodesic
On the largest component of the critical random digraph
Matthew Coulson1 ; Matthew Coulson
1School of Mathematics, University of Birmingham, Edgbaston, Birmingham, UK
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 567-572 
Matthew Coulson; Matthew Coulson. On the largest component of the critical random digraph. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 567-572. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a33/
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     author = {Matthew Coulson and Matthew Coulson},
     title = { On the largest component of the critical random digraph},
     journal = {Acta mathematica Universitatis Comenianae},
     pages = {567--572},
     year = {2019},
     volume = {88},
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     url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a33/}
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Voir la notice de l'article provenant de la source Comenius University

We consider the largest component of the random digraph $D(n,p)$ inside the critical window $p = n^{-1} + \lambda n^{-4/3}$. We show that the largest component $\mathcal{C}_1$ has size of order $n^{1/3}$ in this range. In particular we give explicit bounds on the probabilities that $|\mathcal{C}_1|n^{-1/3}$ is very large or very small that are analogous to those given by Nachmias and Peres for $G(n,p)$.