A local limit theorem for the edge counts of random induced subgraphs of a random graph

TL;DR

This work analyzes the number of edges in the subgraph induced by a random fixed-size vertex set in a dense Erdős–Rényi graph. It develops a two-stage approach: first a central limit theorem for the edge count using a 2D Stein’s method with exchangeable pairs, then a local limit theorem via an iterative smoothing technique that transfers interval probability control to point probabilities. The main results give an explicit mean $KM/N$ and variance $\lambda n^2$ for the limiting Gaussian, and show that the point probabilities of $e(S)$ approximate the Gaussian density $\varphi$ with an error $O(n^{-1/14+\varepsilon})$, uniformly over all integers $z$. The methods apply in the $G_{n,M}$ and $G_{n,p}$ models (with $p\in[\delta,1-\delta]$) and may extend to other subgraph statistics, offering a robust smoothing framework for local limit phenomena in dense random graphs.

Abstract

Consider a `dense' Erdős--Rényi random graph model $G=G_{n,M}$ with $n$ vertices and $M$ edges, where we assume the edge density $M/\binom{n}{2}$ is bounded away from 0 and 1. Fix $k=k(n)$ with $k/n$ bounded away from 0 and~1, and let $S$ be a random subset of size $k$ of the vertices of $G$. We show that with probability $1-\exp(-n^{Ω(1)})$, $G$ satisfies both a central limit theorem and a local limit theorem for the empirical distribution of the edge count $e(G[S])$ of the subgraph of $G$ induced by $S$, where the distribution is over uniform random choices of the $k$-set $S$.

Figures (1)

• Figure 1: Vertex interchange in $G$.

Theorems & Definitions (45)

• Theorem 1
• Theorem 2
• Theorem 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• Remark 7