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Related Experiment Videos

Vacancy localization in the square dimer model.

J Bouttier1, M Bowick, E Guitter

  • 1Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS, CEA/Saclay, 91191 Gif sur Yvette Cedex, France. jeremie.bouttier@cea.fr

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2007
PubMed
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We analyzed a vacancy in the classical dimer model on a square lattice. Vacancies are found to be jammed with a specific probability and exhibit power-law behavior in accessible domain size.

Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Graph Theory

Background:

  • The classical dimer model on a square lattice is a fundamental model in statistical mechanics.
  • Understanding the behavior of defects, such as vacancies, is crucial for characterizing system properties.
  • Previous work established a connection between dimer coverings and spanning trees.

Purpose of the Study:

  • To develop a graph-theoretic framework for classifying dimer configurations with a single vacancy.
  • To investigate the mobility and accessibility of a vacancy within the dimer lattice.
  • To quantify the jamming probability and diffusion characteristics of the vacancy.

Main Methods:

  • Graph-theoretic classification of dimer configurations extending the spanning tree formulation.

Related Experiment Videos

  • Analysis of vacancy motion induced by dimer slidings.
  • Derivation of jamming probabilities and domain size distributions.
  • Numerical simulations with free and periodic boundary conditions.

    • Main Results:

    • A probability of 57/4 - 10√2 for a vacancy to be strictly jammed in an infinite system.
    • Vacancy-accessible domain size distribution follows a power law with exponent 9/8.
    • Probability of a bulk vacancy reaching the boundary in a finite system scales as system size to the power of 1/4.
    • Weak localization of vacancies allows for unbounded diffusion, with a diffusion exponent related to that on spanning trees.

    Conclusions:

    • The developed graph-theoretic formalism provides a powerful tool for studying vacancies in dimer models.
    • Vacancies exhibit jamming and restricted diffusion, characterized by universal power-law behaviors.
    • Numerical simulations confirm the theoretical predictions for vacancy dynamics.