Consider a random $G(n,p)$ graph where $p=\omega(\frac 1n)$, and let $x$ denote the probability that the graph has a connected component of size linear in $n$.

It is well known that $x$ tends to $1$ as $n\to \infty$, even if $p\geq \frac {1+\epsilon}{n}$. However, I am looking for a more exact statement: Something of the form $$x\geq 1-O(f(n,p))$$ where $f$ is some small function of $n$ and $p$.

I'm sure that a reference exists somewhere, but I have yet to find it.

Thanks!