# The critical window in random digraphs

## TL;DR

This paper analyzes the component structure of random directed graphs within the critical window, revealing that the largest component scales as n^{1/3} and providing bounds on its tail probabilities.

## Contribution

It characterizes the size and tail behavior of the largest component in random digraphs during the critical phase, a novel insight into their phase transition.

## Key findings

- Largest component size is of order n^{1/3}
- Explicit tail probability bounds for component size
- Provides detailed understanding of the critical window behavior

## Abstract

We consider the component structure 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 tail probabilities of $|\mathcal{C}_1|n^{-1/3}$.