Mean-Field Monomer-Dimer models. A review

TL;DR

This review summarizes rigorous results on mean-field monomer-dimer models, highlighting Gaussian representations, phase transitions, and non-Gaussian fluctuations at critical points, providing insights into complex statistical mechanics phenomena.

Contribution

It offers a comprehensive overview of analytical solutions, including Gaussian representations and phase transition analysis, for mean-field monomer-dimer models, with new insights into critical behavior.

Findings

Gaussian representation simplifies proofs

Identification of phase coexistence transition

Breakdown of CLT at critical point with non-Gaussian distribution

Abstract

A collection of rigorous results for a class of mean-field monomer-dimer models is presented. It includes a Gaussian representation for the partition function that is shown to considerably simplify the proofs. The solutions of the quenched diluted case and the random monomer case are explained. The presence of the attractive component of the Van der Waals potential is considered and the coexistence phase coexistence transition analysed. In particular the breakdown of the central limit theorem is illustrated at the critical point where a non Gaussian, quartic exponential distribution is found for the number of monomers centered and rescaled with the volume to the power 3/4.