# 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} }

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…

## 9 Citations

### The monopole-dimer model on Cartesian products of plane graphs

- Mathematics
- 2022

. The monopole-dimer model is a signed variant of the monomer-dimer model which has determinantal structure. We extend the monopole-dimer model for planar graphs introduced by the second author…

### Squareness for the Monopole-Dimer Model

- MathematicsAnnals of Combinatorics
- 2020

The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant…

### Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics

- Physics
- 2015

### A Pfaffian Formula for Monomer–Dimer Partition Functions

- MathematicsJournal of Statistical Physics
- 2016

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…

### A Pfaffian Formula for Monomer–Dimer Partition Functions

- Mathematics
- 2015

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…

### Pfaffian Correlation Functions of Planar Dimer Covers

- MathematicsJournal of Statistical Physics
- 2017

The Pfaffian structure of the boundary monomer correlation functions in the dimer-covering planar graph models is rederived through a combinatorial/topological argument. These functions are then…

### Enumeration of Tilings

- Mathematics
- 2015

Note: This survey article will be a chapter in the Handbook of Enumerative Com-binatorics, to be published by CRC Press in early 2015.

### Pfaffian Correlation Functions of Planar Dimer Covers

- Mathematics
- 2016

The Pfaffian structure of the boundary monomer correlation functions in the dimer-covering planar graph models is rederived through a combinatorial/topological argument. These functions are then…

### Squareness for the Monopole-Dimer Model

- Mathematics
- 2016

The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant…

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