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Disordered Monomer-Dimer model on Cylinder graphs

Partha S. Dey, Kesav Krishnan

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.

Disordered Monomer-Dimer model on Cylinder graphs

Abstract

We consider the disordered monomer-dimer model on cylinder graphs , i.e., graphs given by the Cartesian product of the line graph on 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 . 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.

Paper Structure

This paper contains 36 sections, 41 theorems, 232 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Main results
  4. Limit theorems for the free energy
  5. Quenched and Annealed Limit of a typical matching
  6. Height function and Brownian Motion
  7. CLT for the Ground State Energy
  8. Heuristics
  9. Roadmap
  10. Technical Results and Notation
  11. Notation and Assumptions
  12. Free Energy: Mean and Fluctuation Analysis
  13. Mean and Variance Convergence
  14. Proof of Theorem \ref{['thm:Zmeanvar']}
  15. Central Limit Theorem for the Free Energy
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let $Z_n$ be the partition function as defined in def:Z. Assume that $\mathop{\mathrm{\mathds{E}}}\nolimits (\nu_v^{2}+\omega_e^{2})<\infty$. There exist constants $f\in\mathds{R}, \sigma_F\in (0,\infty)$ depending on the distributions of $\omega$ and $\nu$, such that

Figures (2)

  • Figure 1: First step of the subdivision
  • Figure 2: Interlacing shown for the first 5 levels

Theorems & Definitions (70)

  • Definition 1
  • Definition 2: Restricted Partition Function
  • Theorem 1.1: Mean and Variance for the log-partition function
  • Theorem 1.2: Central Limit Theorem for the log-partition function
  • Definition 3
  • Theorem 1.3: Law of Large Numbers for Unpaired Vertices
  • Theorem 1.4: Quenched CLT for $U$
  • Theorem 1.5: Quenched Joint CLT
  • Theorem 1.6: Annealed CLT for $\langle U\rangle_n$
  • Definition 4
  • ...and 60 more