Uniqueness and locality of the ground state of the disordered Monomer-Dimer models on independently weighted Unimodular Bienaym\'e-Galton-Watson trees

Abstract

Consider a finite graph $G=(V(G),E(G))$ and two continuous weight distributions $\omega$ and $\xi$, for which we only assume that $\xi$ is lower bounded. Next, independently draw weights $(w(e))_{e \in E(G)}$ with distribution $\omega$ on edges and $(x(v))_{v \in V(G)}$ with distribution $\xi$ on vertices. The ground state of the monomer-dimer model on the weighted graph $G$ is a collection of edges (dimers) and vertices (monomers) such that every vertex is included in at most one monomer or dimer, and such that the sum of weights on its dimers and monomers is maximised. Take $(G_n,o_n)_{n \in \mathbb{N}}$ to be a sequence of random rooted weighted graphs that converges locally to an independently weighted unimodular Bienaym\'e-Galton-Watson tree $(\mathbb{T},o)$ with vertex-weight distribution $\xi$ and edge-weight distribution $\omega$ . By proving that the ground state of the monomer-dimer model on the tree $(\mathbb{T},o)$ is almost surely unique and locally approximable, we prove that the ground state of the monomer-dimer model on $(G_n,o_n)$ must converge locally to the ground state of the monomer-dimer model on $(\mathbb{T},o)$. This also implies a strong decorrelation property on monomer-dimer models on unimodular Bienaym\'e-Galton-Watson trees.