The critical-window conjecture for giant $j$-components in random hypergraphs

Let $\mathcal{H}^k(n,p)$ be the random $k$-uniform hypergraph with edge probability $p$, and let $j$-components denote components under the adjacency notion used in the paper. Write $\hat{p}_\mathrm{g}$ for the threshold appearing in Theorem~, let $\varepsilon=\varepsilon(n)$, and let $\operatorname{whp}$ mean with high probability. Critical-window conjecture. Theorem~ should hold for all $\varepsilon=\varepsilon(n)$ satisfying

$$\varepsilon\to 0 \qquad\text{and}\qquad \varepsilon^3 n^j\to\infty.$$

Furthermore, if

$$p=(1+cn^{-j/3})\hat{p}_\mathrm{g}$$

for some fixed $c\in\mathbb{R}$, then with high probability all $j$-components have size $O(n^{2j/3})$, and there is more than one $j$-component of $\mathcal{H}^k(n,p)$ of size $\Theta(n^{2j/3})$. The conjecture would sharpen the known lower bound on the width of the critical window for the emergence of a unique largest $j$-component. The surrounding discussion indicates that the existing condition $\varepsilon^2n^{1-\delta}\to\infty$ is probably not best possible for $j\ge 2$; the proposed scale is motivated by the $n^{-j/3}$ critical-window parameter, but no resolution is supplied here.