A Pittel-type limit law for the largest component of random d-uniform hypergraphs: enumeration and excursions
Abstract
Let Hd(n,M) be a d-uniform hypergraph with n vertices and M hyperedges. In the critical window M=n/(d(d−1))(1+λn−1/3)+O(1), its largest component Ln satisfies n−2/3Lnxrightarrowℒ ℱPit((d−1)1/3a,λ(d−1)−2/3). (Here ℱPit(x,μ) 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 Ln. Subsequently, cancellation of the orders n,n2/3,n1/3 in the tree normalization determines the series of rooted hypertrees, then that of unicyclic components and its singular expansions at the radius of convergence ρd. 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 Hd(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−1/3 and evaluate at parameter λh−2/3. The exact Poisson representation dispenses with any separate deconditioning. We also formulate a two-regime d-uniform Bollobas bound (Wright for fixed ℓ, global combinatorial beyond that), and explicitly indicate why neither regime alone suffices to close the analytic route.
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BibTeX
@misc{wirdane2026pitteltype,
title = {{A Pittel-type limit law for the largest component of random $d$-uniform hypergraphs: enumeration and excursions}},
author = {Youssouf Wirdane},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27276},
eprint = {2610.27276},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27276},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Pittel-type limit law for the largest component of random d-uniform hypergraphs: enumeration and excursions
AU - Youssouf Wirdane
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27276
UR - https://arxiv.org/abs/2610.27276
AB - Let H_d(n,M) be a d-uniform hypergraph with n vertices and M hyperedges. In the critical window M=n/(d(d−1))(1+λn^−1/3)+O(1), its largest component L_n satisfies n^−2/3L_nxrightarrowℒ ℱ_Pit((d−1)^1/3a,λ(d−1)^−2/3). (Here ℱ_Pit(x,μ) 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^2/3,n^1/3 in the tree normalization determines the series of rooted hypertrees, then that of unicyclic components and its singular expansions at the radius of convergence ρ_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^−1/3 and evaluate at parameter λh^−2/3. The exact Poisson representation dispenses with any separate deconditioning. We also formulate a two-regime d-uniform Bollobas bound (Wright for fixed ℓ, global combinatorial beyond that), and explicitly indicate why neither regime alone suffices to close the analytic route.
ER -