Disordered Monomer-Dimer model on Cylinder graphs

TL;DR

This paper studies a disordered monomer-dimer model on cylinder graphs, proving convergence of free energy, laws of large numbers, central limit theorems, and Brownian motion limits for associated height functions.

Contribution

It introduces a new analysis of the disordered monomer-dimer model on cylinder graphs, establishing limit theorems and convergence results for free energy, monomer counts, and height functions.

Findings

Free energy converges to a limit.

Number of monomers satisfies LLN and CLT.

Height function converges to a limiting function and exhibits Brownian motion limits.

Abstract

We consider the disordered monomer-dimer model on cylinder graphs $\mathcal{G}_n$, i.e., graphs given by the Cartesian product of the line graph on $n$ vertices, and a deterministic graph. The edges carry i.i.d. random weights, and the vertices also carry i.i.d. random weights, not necessarily from the same distribution. Given the random weights, we define a Gibbs measure on the space of monomer-dimer configurations on $\mathcal{G}_n$. We show that the associated free energy converges to a limit, and with suitable scaling and centering, satisfies a central limit theorem. We also show that the number of monomers in a typical configuration satisfies a law of large numbers and a central limit theorem with appropriate centering and scaling. Finally, for an appropriate height function associated with a matching, we show convergence to a limiting function and prove the Brownian motion limit about the limiting height function in the sense of finite-dimensional distributions.