May 2021 Anti-concentration for subgraph counts in random graphs
Jacob Fox, Matthew Kwan, Lisa Sauermann
Author Affiliations +
Ann. Probab. 49(3): 1515-1553 (May 2021). DOI: 10.1214/20-AOP1490


Fix a graph H and some p(0,1), and let XH be the number of copies of H in a random graph G(n,p). Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of XH, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that XH falls in some small interval or is equal to some particular value? In this paper, we prove the almost-optimal result that if H is connected then for any xN we have Pr(XH=x)n1v(H)+o(1). Our proof proceeds by iteratively breaking XH into different components which fluctuate at “different scales”, and relies on a new anti-concentration inequality for random vectors that behave “almost linearly.”


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Jacob Fox. Matthew Kwan. Lisa Sauermann. "Anti-concentration for subgraph counts in random graphs." Ann. Probab. 49 (3) 1515 - 1553, May 2021.


Received: 1 May 2019; Revised: 1 September 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1490

Primary: 05C80 , 60C05

Keywords: anti-concentration , Random graphs

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 3 • May 2021
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