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

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 ${\ell }^1$ 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.