A Pfaffian formula for the monomer–dimer model on surface graphs

@article{Pham2017APF,
  title={A Pfaffian formula for the monomer–dimer model on surface graphs},
  author={Anh Minh Pham},
  journal={Letters in Mathematical Physics},
  year={2017},
  volume={108},
  pages={1885-1904}
}
  • A. Pham
  • Published 2 May 2017
  • Mathematics
  • Letters in Mathematical Physics
AbstractWe consider the monomer–dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani et al. (J Stat Phys 163(2):211–238, 2016) for graphs embedded in the disc. Our proof is based on an extension of a bijective method mentioned in Giuliani et al. (2016), together with the Pfaffian formula… 

Figures from this paper

The dimer and Ising models on non-orientable surfaces
Le but de cette these est d'etudier le modele des dimeres, le modele des monomeres-dimeres, et le modele d'Ising, plus particulierement les fonctions de partition de ces modeles pour des graphes

References

SHOWING 1-10 OF 22 REFERENCES
A Pfaffian Formula for Monomer–Dimer Partition Functions
We consider the monomer–dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights, with the restriction that the only non-zero monomer weights are those on
Dimers on Graphs in Non-Orientable Surfaces
The main result of this article is a Pfaffian formula for the partition function of the dimer model on any graph Γ embedded in a closed, possibly non-orientable surface Σ. This formula is suitable
Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary.
  • F. Y. Wu
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
TLDR
The mathematical content of the Tzeng-Wu solution for a single monomer on the boundary by evaluating a Pfaffian is clarified by identifying it as the product of the nonzero eigenvalues of the Kasteleyn matrix.
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
Exact solution of a monomer-dimer problem: a single boundary monomer on a nonbipartite lattice.
TLDR
This paper derives the solution of the monomer-dimer problem on a nonbipartite lattice by mapping the problem onto one of closed-packed dimers on a related lattice from asymptotic expansions of the free energy that the central charge in the logarithmic conformal field theory assumes the value c=-2.
Boundary monomers in the dimer model.
TLDR
The equivalence of the 2n -point correlation functions with those of a complex free fermion is proved, thereby reinforcing the description of the monomer-dimer model by a conformal free-field theory with central charge c=1.
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
Matchings in Graphs on Non-orientable Surfaces
  • G. Tesler
  • Mathematics
    J. Comb. Theory, Ser. B
  • 2000
TLDR
“Crossing orientations,” the analogue of Kasteleyn's “admissible orientations” in this context, are introduced, describing how the Pfaffian of a signed adjacency matrix of a graph gives the sign of each perfect matching according to the number of edge-crossings in the matching.
Dimers on Surface Graphs and Spin Structures. I
Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of
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