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*}

$\mathcal{C}_{\max}$

$\lambda$

We investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework for studying the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.

The largest components of the critical Erdős–Rényi graph, G(n, p) with p = 1 / n, have size of order n 2/3 with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size an 2/3 for large a. Our results, which extend the work of Pittel (2001), allow a to depend upon n and also hold for a range of values of p around 1 / n. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.

In this work we give precise asymptotic expressions for the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.