Central Limit Theorems for global and local empirical measures of diffusions on Erd\H{o}s-R\'enyi graphs

TL;DR

This paper establishes Central Limit Theorems for empirical measures of diffusions on Erdős-Rényi graphs, highlighting the impact of initial conditions and using advanced inequalities for proofs.

Contribution

It extends CLT results to diffusions on diluted Erdős-Rényi graphs, analyzing the effect of initial conditions on fluctuation universality.

Findings

Fluctuations match mean-field results with independent initial conditions.

Non-universal fluctuations occur with graph-dependent initial data.

Extensions of Grothendieck inequality are key to the proofs.

Abstract

We address the issue of the Central Limit Theorem for (both local and global) empirical measures of diffusions interacting on a possibly diluted Erd\H{o}s-R\'enyi graph. Special attention is given to the influence of initial condition (not necessarily i.i.d.) on the nature of the limiting fluctuations. We prove in particular that the fluctuations remain the same as in the mean-field framework when the initial condition is chosen independently from the graph. We give an example of non-universal fluctuations for carefully chosen initial data that depends on the graph. A crucial tool for the proof is the use of extensions of Grothendieck inequality.