Exact finite-size corrections in the dimer model on a cylinder

Abstract The exact finite-size corrections to the free energy F of the dimer model on lattice <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mo>×</mml:mo> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">N</mml:mi> </mml:math> with cylindrical boundary conditions have been derived for three cases where the lattice is completely covered by dimers: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:math> ; <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> </mml:math> ; and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:math> . For these types of cylinders, ratios r p ( ρ ) of the p th coefficient of F have been calculated for the infinitely long cylinder ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> ) and infinitely long strip ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">N</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:math> ) at varying aspect ratios. As in previous studies of the dimer model on the rectangular lattice with free boundary conditions and for the Ising model with Brascamp-Kunz boundary conditions, the limiting values p → ∞ exhibit abrupt anomalous behaviour of ratios r p ( ρ ) at certain values of ρ . These critical values of ρ and the limiting values of the finite-size expansion coefficient ratios vary between the different models.