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

TL;DR

This paper establishes a local limit theorem for the distribution of the number of r-cliques in Erdős–Rényi random graphs, providing precise probabilistic bounds using novel characteristic function techniques.

Contribution

It introduces a new method for bounding characteristic functions of degree-r polynomials in Bernoulli variables, enabling the proof of the local limit theorem.

Findings

Proves a local limit theorem for r-cliques in G(n,p)

Provides bounds in both ℓ∞ and ℓ¹ metrics

Develops a new technique for characteristic function estimation

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.