Limit Theorems in the Imitative Monomer-Dimer Mean-Field Model via Stein's Method

TL;DR

This paper analyzes the fluctuation behavior of the monomer density in the imitative monomer-dimer model on complete graphs, establishing limit theorems and phase transition properties using Stein's method.

Contribution

It provides a full characterization of the fluctuation limits of monomer density across all parameters, including at criticality, via Stein's method.

Findings

Normal distribution convergence away from criticality

Non-normal limit at criticality with exponent 3/4

Conditional CLTs along the critical line

Abstract

We consider the imitative monomer-dimer model on the complete graph introduced in [1]. It was understood that this model is described by the monomer density and has a phase transition along certain critical line. By reverting the model to a weighted Curie-Weiss model with hard core interaction, we establish the complete description of the fluctuation properties of the monomer density on the full parameter space via Stein's method of exchangeable pairs. We show that this quantity exhibits the central limit theorem away from the critical line and enjoys a non-normal limit theorem at criticality with normalized exponent $3/4$. Furthermore, our approach also allows to obtain the conditional central limit theorems along the critical line. In all these results, the Berry-Esseen inequalities for the Kolomogorov-Smirnov distance are given.