Upper tails for homomorphism counts in sparse random hypergraphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918151722106880 |
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| author | Cook, Nicholas A. Nguyen, Nguyen |
| author_facet | Cook, Nicholas A. Nguyen, Nguyen |
| contents | The "infamous upper tail problem" for $r$-uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$-uniform hypergraph $\boldsymbol{G}$ exceeds its expectation by a constant factor. The problem was popularized by Janson and Ruciński and, particularly in the case of graphs ($r=2$), has been a driving example in the development of nonlinear large deviations theory. Recent work of the first author with Dembo and Pham has accomplished the \emph{naive mean-field reduction step}, reducing the upper tail problem to an entropic variational problem on a space of weighted graphs. The latter was resolved for counts of $r$-uniform cliques and a certain linear 3-uniform hypergraph by Liu and Zhao, who also conjectured a general formula. We confirm their conjecture for other classes of hypergraphs, including complete $r$-partite $r$-graphs, tight cycles, and the Fano plane. We also prove a general large deviation upper bound for counts of $r$-graphs $H$ satisfying certain edge covering properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Upper tails for homomorphism counts in sparse random hypergraphs Cook, Nicholas A. Nguyen, Nguyen Combinatorics Probability 60F10, 60C05, 60B20, 05C65 The "infamous upper tail problem" for $r$-uniform hypergraphs is to estimate the probability that the number of copies of a fixed hypergraph $H$ in a large binomial $r$-uniform hypergraph $\boldsymbol{G}$ exceeds its expectation by a constant factor. The problem was popularized by Janson and Ruciński and, particularly in the case of graphs ($r=2$), has been a driving example in the development of nonlinear large deviations theory. Recent work of the first author with Dembo and Pham has accomplished the \emph{naive mean-field reduction step}, reducing the upper tail problem to an entropic variational problem on a space of weighted graphs. The latter was resolved for counts of $r$-uniform cliques and a certain linear 3-uniform hypergraph by Liu and Zhao, who also conjectured a general formula. We confirm their conjecture for other classes of hypergraphs, including complete $r$-partite $r$-graphs, tight cycles, and the Fano plane. We also prove a general large deviation upper bound for counts of $r$-graphs $H$ satisfying certain edge covering properties. |
| title | Upper tails for homomorphism counts in sparse random hypergraphs |
| topic | Combinatorics Probability 60F10, 60C05, 60B20, 05C65 |
| url | https://arxiv.org/abs/2509.26569 |