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Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density.

Yong Kong1

  • 1Department of Mathematics, National University of Singapore, Singapore 117543. matky@nus.edu.sg

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2007
PubMed
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Investigating dimer arrangements on 2D lattices, this study reveals a universal logarithmic correction in free energy. This finding, explained by new asymptotic theory, offers precise monomer-dimer constants for rectangular lattices.

Area of Science:

  • Statistical Mechanics
  • Computational Physics
  • Combinatorics

Background:

  • The monomer-dimer model on 2D lattices is computationally intractable (#P-complete).
  • Understanding dimer arrangements is crucial for various physical systems.

Purpose of the Study:

  • To investigate dimer arrangements on mxn rectangular lattice strips with fixed dimer density.
  • To identify and characterize finite-size corrections to the free energy.

Main Methods:

  • Exact computational methods were employed.
  • Analysis of free energy per lattice site for varying dimer densities.
  • Application of Pemantle and Wilson's asymptotic theory.

Main Results:

  • A logarithmic correction term (-1/2) was found in the finite-size correction of free energy for any dimer density (0 < rho < 1).

Related Experiment Videos

  • Highly accurate free energy values were obtained for large lattices, with the monomer-dimer constant determined to 11 decimal places.
  • Dependence on lattice width parity was observed for finite widths near close packing.

    • Conclusions:

    • The study provides a precise understanding of free energy behavior in 2D lattice dimer models.
    • The findings validate new asymptotic theories and offer accurate constants for physical models.
    • The research contributes to the computational and statistical analysis of complex lattice systems.