Grassmannian representation of the two-dimensional monomer-dimer model

Abstract

We present an application of the Grassmann algebra to the problem of the monomer-dimer statistics on a two-dimensional square lattice. The exact partition function, or total number of possible configurations, of a system of dimers with a finite set of n monomers with fixed positions can be expressed via a quadratic fermionic theory. We give an answer in terms of a product of two pfaffians and the solution is closely related to the Kasteleyn result of the pure dimer problem. Correlation functions are in agreement with previous results, both for monomers on the boundary, where a simple exact expression is available in the discrete and continuous case, and in the bulk where the expression is evaluated numerically.

7 pages. Accepted for publication in Physical Review E

References in corpus (4)

• Return probability for the loop-erased random walk and mean height in sandpile : a proof
• The Pfaffian solution of a dimer-monomer problem: Single monomer on the boundary
• Boundary monomers in the dimer model
• Vacancy localization in the square dimer model

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• Integrability and conformal data of the dimer model
• Exact finite-size corrections in the dimer model on a planar square lattice
• A Statistical Model of Current Loops and Magnetic Monopoles
• Pfaffian Correlation Functions of Planar Dimer Covers
• The expectation value of the number of loops and the left-passage probability in the double-dimer model
• Exact finite-size corrections in the dimer model on a cylinder