A Pittel-type limit law for the largest component of random $d$-uniform hypergraphs: enumeration and excursions

Let $H_d(n,M)$ be a $d$-uniform hypergraph with $n$ vertices and $M$ hyperedges. In the critical window $M=\frac{n}{d(d-1)}(1+λn^{-1/3})+O(1)$, its largest component $L_n$ satisfies $$n^{-2/3}L_n\xrightarrow{\mathcal L} \mathcal{F}_{\mathrm{Pit}}\bigl((d-1)^{1/3}a,λ(d-1)^{-2/3}\bigr).$$ (Here $\math