I've been looking for a short proof to show the phase transition for the largest connected component in an Erdős-Rényi graph.
All the papers I read were showing this result with a lot of precision (I mean, they were looking for the best boundary of the size of the largest component in multiple regime). The first problem I have with this is that I need to write a proof on less than 2 pages. The second is that it's difficult to understand their proofs because there's too many notations.
What I just want to do is to show that the size of the biggest component is small for $p<1/n$, much bigger for $p>1/n$ (and maybe showing there is a unique component for a certain $p$).
Do you know any simple proof to show this result?
For both proofs you're going to need to bound by a branching process in some way, and that's going to be harder for the supercritical case.
If you just want a crude bound that all components are sublinear with high probability for $p=(1-\delta)/n$, that's relatively easy. Fix a vertex $v$ and write $C_v$ for the component it's in. We can bound the average number of vertices at distance $k$ from $v$ by $(1-\delta)^k$ (by induction), so $E(|C-v|)\leq\sum_{k=0}^{n-1}(1-\delta)^k<\delta^{-1}$.
Now what is $P(\text{largest component}\geq\varepsilon n)$? If the largest component contains at least $\varepsilon n$ vertices then with probability at least $\varepsilon$ one of them is $v$. So we have $$P(|C_v|\geq \varepsilon n)\geq \varepsilon P(\text{largest component }\geq\varepsilon n).$$ But by Markov's inequality $$P(|C_v|\geq \varepsilon n)\leq \frac{E(|C_v|)}{\varepsilon n},$$ so we get $$P(\text{largest component}\geq\varepsilon n)\leq \frac1{\delta\varepsilon^2n}.$$