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

Abstract

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.

An elementary approach to component sizes in critical random graphs

“…Our claim that the approach introduced in [9] is robust and that Proposition 1 leads to simple upper bounds for in several models of random graphs at criticality is justified in Sections 2.1, 2.2, and 2.3 below, where we use Proposition 1 to obtain polynomial upper bounds for the above probability in three particular models of random graphs.…”

“…In order to better understand the statement of our main result (Theorem 1 below), we first need to recall the definition of an exploration process, which is an algorithmic procedure used to reveal the components of a given graph; see e.g. [9], [22], [23], [27], and references therein. As we will see in a moment, when the graph under investigation is random such an exploration process reduces the study of component sizes to the analysis of the trajectory of a random process, which looks like (but is not quite) a random walk.…”

“…Remark 1. We remark that our description of an exploration process is slightly different from those provided in [9], [23], and [27], for example. Indeed, in our setting the algorithm is actually run only in the case where V n is not an isolated vertex, and the exploration starts from two active vertices and not from one active vertex, as usually happens.…”

“…In [9] De Ambroggio and Roberts introduced a ballot-type result (Lemma 1 below) to provide a new, purely probabilistic proof of the fact that in the model considered in the so-called critical window , i.e. when p is of the form , the probability of observing a maximal cluster of size larger than tends to zero (as ) exponentially fast in A .…”

“…The purpose of this work is to show that part of the argument used in [9] to prove the upper bound in (1) is quite general and can be used to obtain, in a surprisingly simple way, polynomial upper bounds for in different models of random graphs when considered at criticality. Specifically, we apply our method to three different models, namely a model of a random intersection graph, a random graph obtained through p -bond percolation on a general d -regular graph, and a model of an inhomogeneous random graph, and we show that is unlikely to be much larger than in these models.…”