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Dimer statistics on a Bethe lattice.

A B Harris¹, Michael Cohen

¹Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA. harris@physics.upenn.edu

The Journal of Chemical Physics

|November 23, 2006

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We present exact solutions for dimer coverings on Bethe lattices, including interactions and anisotropy. A surprising link between maximal dimer coverage and branched polymers is revealed.

Area of Science:

Statistical Mechanics
Condensed Matter Physics
Mathematical Physics

Background:

Dimer coverings are fundamental in statistical mechanics.
Bethe lattices offer a tractable model for studying complex systems.
Understanding dimer statistics is crucial for various physical phenomena.

Purpose of the Study:

To derive exact solutions for dimer coverings on Bethe lattices under various conditions.
To explore the impact of interactions, anisotropy, and random potentials on dimer statistics.
To establish connections between dimer models and other statistical systems like branched polymers.

Main Methods:

Utilizing a simple geometrical argument for noninteracting dimers.
Employing a general algebraic formulation for lattice statistical problems.

Related Experiment Videos

Analyzing dimer statistics in the presence of short-range interactions, anisotropy, and quenched random potentials.

Main Results:

Reproduced the exact result for noninteracting hard-core dimers.
Obtained exact solutions for interacting and anisotropic dimer models on Bethe lattices.
Found an identity between the partition function for maximal dimer coverage and branched polymer statistics.
Derived an exact solution for residual vacancies in a one-dimensional dimer deposition model.

Conclusions:

The algebraic formulation provides a powerful tool for analyzing lattice statistical problems, including loop corrections.
Dimer statistics on Bethe lattices exhibit rich behavior under various perturbations.
The connection to branched polymers highlights unexpected universality in statistical models.
Exact solutions offer valuable insights into the behavior of physical systems with constrained configurations.