An Erdős-Rényi graph is a random graph, selected according to the distribution obtained one where we have some number $n$ of nodes, and some probability $p$ of each potential edge being present. One often asks questions of this distribution when $p$ is taken as a function of $n$, and we take the limit as $n \to \infty$. Classic results on properties of the graph which hold almost surely, from Erdős and Rényi's original paper on random graphs, are as follows:
In the case that $p = (1 - \Omega(1)) \cdot \tfrac1n$, the graph consists almost entirely of trees of size $O(\log n)$, and a finite number of components which contain a single cycle.
In the case that $p = (1 + \Omega(1)) \cdot \tfrac1n$, the graph consists of a single "giant" component which contains a positive fraction of all the vertices and has multiple cycles, together with a finite number of components which contain a single cycle and trees of size $O(\log n)$.
It is clear from this that between $p = n^{-1} \pm \Theta(n^{-1})$, the probability that the graph contains a component which has multiple cycles goes from 0 to 1. Furthermore, for $p = \tfrac1n$ precisely, we expect the number of vertices which belong to a component which has a single cycle also to scale as $\Theta(n^{2/3})$: connecting any two of these vertices (even in the same component) will result in a component with two cycles, so for $n^{-1} + \omega(n^{-4/3})$, we should already expect to see components with multiple cycles.
What I'm interested is: what is known about the growth of functions $f(n)$ for which $p = n^{-1} + \Theta(f(n))$ implies that there is almost surely a component with multiple cycles? How much better can we do than $f(n) \in \omega(n^{-4/3})$?
For instance: by Theorem 8a of Erdős and Rényi's original paper on the evolution of random graphs, we would expect an increasing probability, bounded only by zero and one, of having a single cycle with at least one diagonal edge, for $p = \tfrac{1}{n}(1 + c\tfrac{1}{\sqrt n})$, for any constant $c$; this would show that any $f(n) \in \omega(n^{-3/2})$ would suffice. However, a problem with this Theorem was discovered by Łuczak and Wierman, so such a threshold does not seem to hold. It would also be unnecessarily restrictive even if true, as it would describe the special case of a component which has two cycles which overlap on an edge: one could hope to show the existence (for instance) of components with two cycles which may be attached to distant leaves of a large tree, with a smaller $f$.
How small a function $f$ can we find a transition from every component being a tree or unicyclic, to there being components with multiple cycles?
