Exact Solution of the Dimer Model on the Square and Triangular Lattice

In this thesis we give an exact solution of the dimer model on the square and triangular lattice with different boundary conditions. Concretely, we being by discussing Kasteleyn’s contribution to the dimer model studies. We then prove the Pfaÿan Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifcally, we prove that the Pfaffan of the Kasteleyn periodic-periodic matrix is negative, while the Pfaÿans of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. Finally, we obtain the full asymptotic expansion of the partition function of the two-dimensional dimer model on the m × n square lattice with free and periodic boundary conditions. We show that the asymptotic expansion goes over powers of S−1, where S =(m + 1)(n + 1) for the free boundary conditions, and S = mn for the periodic boundary conditions. The coefficients of the asymptotic expansion are expressed in terms of the Kronecker double series, the Dedekind eta function, and the Jacobi theta functions. Furthermore, as an application of the Pfaÿan Sign Theorem, we obtain an asymptotic behavior with an exponentially small error term of the partition function of the two-dimensional dimer model on the m × n triangular lattice on the torus