Anticoncentration for subgraph statistics

Kwan MA, Sudakov B, Tran T. 2019. Anticoncentration for subgraph statistics. Journal of the London Mathematical Society. 99(3), 757–777.


Journal Article | Published | English

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Author
Kwan, Matthew AlanIST Austria; Sudakov, Benny; Tran, Tuan
Abstract
Consider integers π‘˜,β„“ such that 0β©½β„“β©½(π‘˜2) . Given a large graph 𝐺 , what is the fraction of π‘˜ -vertex subsets of 𝐺 which span exactly β„“ edges? When 𝐺 is empty or complete, and β„“ is zero or (π‘˜2) , this fraction can be exactly 1. On the other hand, if β„“ is far from these extreme values, one might expect that this fraction is substantially smaller than 1. This was recently proved by Alon, Hefetz, Krivelevich, and Tyomkyn who initiated the systematic study of this question and proposed several natural conjectures. Let β„“βˆ—=min{β„“,(π‘˜2)βˆ’β„“} . Our main result is that for any π‘˜ and β„“ , the fraction of π‘˜ -vertex subsets that span β„“ edges is at most log𝑂(1)(β„“βˆ—/π‘˜)√ π‘˜/β„“βˆ—, which is best-possible up to the logarithmic factor. This improves on multiple results of Alon, Hefetz, Krivelevich, and Tyomkyn, and resolves one of their conjectures. In addition, we also make some first steps towards some analogous questions for hypergraphs. Our proofs involve some Ramsey-type arguments, and a number of different probabilistic tools, such as polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a β€˜slice’ of the Boolean hypercube.
Publishing Year
Date Published
2019-05-03
Journal Title
Journal of the London Mathematical Society
Volume
99
Issue
3
Page
757-777
ISSN
eISSN
IST-REx-ID

Cite this

Kwan MA, Sudakov B, Tran T. Anticoncentration for subgraph statistics. Journal of the London Mathematical Society. 2019;99(3):757-777. doi:10.1112/jlms.12192
Kwan, M. A., Sudakov, B., & Tran, T. (2019). Anticoncentration for subgraph statistics. Journal of the London Mathematical Society. Wiley. https://doi.org/10.1112/jlms.12192
Kwan, Matthew Alan, Benny Sudakov, and Tuan Tran. β€œAnticoncentration for Subgraph Statistics.” Journal of the London Mathematical Society. Wiley, 2019. https://doi.org/10.1112/jlms.12192.
M. A. Kwan, B. Sudakov, and T. Tran, β€œAnticoncentration for subgraph statistics,” Journal of the London Mathematical Society, vol. 99, no. 3. Wiley, pp. 757–777, 2019.
Kwan MA, Sudakov B, Tran T. 2019. Anticoncentration for subgraph statistics. Journal of the London Mathematical Society. 99(3), 757–777.
Kwan, Matthew Alan, et al. β€œAnticoncentration for Subgraph Statistics.” Journal of the London Mathematical Society, vol. 99, no. 3, Wiley, 2019, pp. 757–77, doi:10.1112/jlms.12192.
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