Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness and locality of the ground state of the disordered Monomer-Dimer models on independently weighted Unimodular Bienaymé-Galton-Watson trees

Mihyun Kang, Mike Liu

Abstract

Consider a finite graph $G=(V(G),E(G))$ and two continuous weight distributions $ω$ and $ξ$, for which we only assume that $ξ$ is lower bounded. Next, independently draw weights $(w(e))_{e \in E(G)}$ with distribution $ω$ on edges and $(x(v))_{v \in V(G)}$ with distribution $ξ$ 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 Bienaymé-Galton-Watson tree $(\mathbb{T},o)$ with vertex-weight distribution $ξ$ and edge-weight distribution $ω$ . 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 Bienaymé-Galton-Watson trees.

Uniqueness and locality of the ground state of the disordered Monomer-Dimer models on independently weighted Unimodular Bienaymé-Galton-Watson trees

Abstract

Consider a finite graph and two continuous weight distributions and , for which we only assume that is lower bounded. Next, independently draw weights with distribution on edges and with distribution on vertices. The ground state of the monomer-dimer model on the weighted graph 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 to be a sequence of random rooted weighted graphs that converges locally to an independently weighted unimodular Bienaymé-Galton-Watson tree with vertex-weight distribution and edge-weight distribution . By proving that the ground state of the monomer-dimer model on the tree is almost surely unique and locally approximable, we prove that the ground state of the monomer-dimer model on must converge locally to the ground state of the monomer-dimer model on . This also implies a strong decorrelation property on monomer-dimer models on unimodular Bienaymé-Galton-Watson trees.
Paper Structure (9 sections, 7 theorems, 73 equations, 3 figures)

This paper contains 9 sections, 7 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Unimodular Bienaymé-Galton-Watson trees
  5. Message-passing for monomer-dimer ground states
  6. Uniqueness of messages on the limit tree and convergence
  7. Invariance trick
  8. Case $\xi$ continuous
  9. Case $\xi$ with a singular atom at its minimum

Key Result

Theorem 1

Let $(\mathbb{T},w,x,o)$ be a unimodular Bienaymé-Galton-Watson tree with reproduction law $\pi$ with vertex weights $(x(v))_{v \in V(\mathbb{T})}$ with distribution $\xi$ and edge weights $(w(e))_{e \in E(\mathbb{T})}$ with distribution $\omega$. Assume that: Then there almost surely exists a unique optimal monomeric-dimeric configuration $\mathbb{MD}$ on $(\mathbb{T},w,x,o)$ in the following se

Figures (3)

  • Figure 1: A $3-$neighbourhood of a vertex-rooted UBGW tree with the law $\pi$ of the number of children drawn on the root $o$ and the law $\hat{\pi}$ of the number of children on every non-root vertex.
  • Figure 2: The message $Z(u,v)$ is defined as the difference between the weight of the optimal configuration in the tree in the box and the dotted tree, which is equivalent to the gain of attaching $v$ to the dotted forest through the dashed edges.
  • Figure 3: Deduction of the recursion satisfied by the messages.

Theorems & Definitions (21)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1.1
  • Corollary 2
  • Remark 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 11 more