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 $M=\frac{n}{d(d-1)}(1+\lambda n^{-1\mathrm{/}3})+O(1)$, its largest component $L_n$ satisfies $$n^{-2\mathrm{/}3}{L}_n\stackrel{\mathcal{L}}{\to }{\mathcal{F}}_{\mathrm{P}\mathrm{i}\mathrm{t}}((d-1{)}^{1\mathrm{/}3}a,\lambda (d-1{)}^{-2\mathrm{/}3})\mathrm{.}$$ (Here ${\mathcal{F}}_{\mathrm{P}\mathrm{i}\mathrm{t}}(x,\mu )$ denotes the limiting law of the largest component of the critical random graph established by Pittel (2001).) The proof rests on two pillars. First, a bivariate exponential formula gives the exact finite-size law of $L_n$. Subsequently, cancellation of the orders $n,n^{2\mathrm{/}3},n^{1\mathrm{/}3}$ in the tree normalization determines the series of rooted hypertrees, then that of unicyclic components and its singular expansions at the radius of convergence ${\rho }_d$. These calculations are exact; they serve the fixed-excess identities and the cancellation mechanism, and nothing more: no unproved uniformity in the excess enters the proof of the limit law. Next, by exploration and martingale approximation, the ordered sizes converge unconditionally to the excursion lengths of a Brownian motion with parabolic drift. A monotone coupling transfers the law to $H_d(n,M)$. The limit law is then identified with Pittel’s law by a Brownian rescaling: if $h=d-1$, it suffices to multiply lengths by $h^{-1\mathrm{/}3}$ and evaluate at parameter $\lambda h^{-2\mathrm{/}3}$. The exact Poisson representation dispenses with any separate deconditioning. We also formulate a two-regime $d$-uniform Bollobas bound (Wright for fixed $\mathrm{\ell }$, global combinatorial beyond that), and explicitly indicate why neither regime alone suffices to close the analytic route.