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A Local Limit Theorem for Cliques in G(n,p) (1811.03527v1)

Published 8 Nov 2018 in math.CO

Abstract: We prove a local limit theorem the number of $r$-cliques in $G(n,p)$ for $p\in(0,1)$ and $r\ge 3$ fixed constants. Our bounds hold in both the $\ell\infty$ and $\ell1$ metric. The main work of the paper is an estimate for the characteristic function of this random variable. This is accomplished by introducing a new technique for bounding the characteristic function of constant degree polynomials in independent Bernoulli random variables, combined with a decoupling argument.

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