A Statistical Model of Current Loops and Magnetic Monopoles

@article{Ayyer2013ASM,
  title={A Statistical Model of Current Loops and Magnetic Monopoles},
  author={Arvind Ayyer},
  journal={Mathematical Physics, Analysis and Geometry},
  year={2013},
  volume={18},
  pages={1-19}
}
  • A. Ayyer
  • Published 23 November 2013
  • Mathematics
  • Mathematical Physics, Analysis and Geometry
We formulate a natural model of loops and isolated vertices for arbitrary planar graphs, which we call the monopole-dimer model. We show that the partition function of this model can be expressed as a determinant. We then extend the method of Kasteleyn and Temperley-Fisher to calculate the partition function exactly in the case of rectangular grids. This partition function turns out to be a square of a polynomial with positive integer coefficients when the grid lengths are even. Finally, we… 

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References

SHOWING 1-10 OF 42 REFERENCES

Grassmannian representation of the two-dimensional monomer-dimer model.

  • N. AllegraJ. Fortin
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2014
An application of the Grassmann algebra to the problem of the monomer-dimer statistics on a two-dimensional square lattice is presented and the solution is closely related to the Kasteleyn result of the pure dimer problem.

The Signed Loop Approach to the Ising Model: Foundations and Critical Point

The signed loop approach is a beautiful way to rigorously study the two-dimensional Ising model with no external field. In this paper, we explore the foundations of the method, including details that

Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries.

  • Y. Kong
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
Using exact computations, a logarithmic correction term is found in the finite-size correction of the free energy per lattice site of classical hard-core monomer-dimer models on m x n plane lattice strips with free boundaries for an arbitrary number v of monomers.

Dimer Statistics and Phase Transitions

After the introduction of the concept of lattice graph and a brief discussion of its role in the theory of the Ising model, a related combinatorial problem is discussed, namely that of the statistics

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has

Vacancy localization in the square dimer model.

The classical dimer model on a square lattice with a single vacancy is studied by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers, and a probability is found for the vacancy to be strictly jammed in an infinite system.

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

  • Y. Kong
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
The exact computational method is used to investigate the number of ways to arrange dimers on mxn two-dimensional rectangular lattice strips with fixed dimer density rho to find a logarithmic correction term in the finite-size correction of the free energy per lattice site.

Local statistics of lattice dimers

The emergence of the electrostatic field as a Feynman sum in random tilings with holes

We consider random lozenge tilings on the triangular lattice with holes $Q_1,...,Q_n$ in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge

Statistical Mechanics of Dimers on a Plane Lattice. II. Dimer Correlations and Monomers

In part I of this paper, exact expressions were obtained for the partition function and thermodynamic properties of an $m\ifmmode\times\else\texttimes\fi{}n$ plane square lattice filled with