Subcritical largest-component asymptotics near the hypercube scaling window

Let $p_c(n)$ be the critical probability for the random subgraph of the $n$-cube, let $V=2^n$, and write

$$p=p_c(n)-\epsilon n^{-1}.$$

Assume $\epsilon\rightarrow0$ and $\epsilon e^{\delta n}\rightarrow\infty$ for every $\delta>0$. Let ${\mathcal C}_{\rm \max}$ be the largest component. Subcritical largest-component asymptotics conjecture. Almost surely,

$$|{\mathcal C}_{\rm \max}|=\frac{2\log V}{\epsilon^2}[1+o(1)].$$

If instead $\epsilon\rightarrow0$ and $\epsilon 2^{n/3}\rightarrow\infty$, then almost surely

$$|{\mathcal C}_{\rm \max}|=\Theta\left(\frac{2\log(\epsilon^3V)}{\epsilon^2}\right).$$

These assertions extend the known subcritical estimates toward and beyond the scaling window, whose width is of order $V^{-1/3}$.