Central limit theorem for statistics of subcritical configuration models

TL;DR

This paper proves a central limit theorem for additive statistics in sub-critical configuration models, requiring certain moment, variance, and degree sequence conditions, with applications to well-known graph statistics.

Contribution

It establishes CLTs for a broad class of statistics in sub-critical configuration models under minimal assumptions, extending previous results.

Findings

CLT holds for component counts, log-partition function, maximum cut-size

Assumptions reduce to linear variance growth for bounded degree sequences

Proof uses martingale-difference arrays and exploration process

Abstract

We consider sub-critical configuration models and show that the central limit theorem for any additive statistic holds when the statistics satisfies a fourth moment assumption, a variance lower bound and the degree sequence of graph satisfies a growth condition. If the degree sequence is bounded, for well known statistics like component counts, log-partition function, and maximum cut-size which are Lipschitz under addition of an edge or switchings then the assumptions reduce to linear growth condition for the variance of the statistic. Our proof is based on an application of the central limit theorem for martingale-difference arrays due to Mcleish (1974) to a suitable exploration process.