Finite-size effect for the critical point of an anisotropic dimer model of domain walls

The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable τ and a shape factor scrr=$N^2$/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable τ is given by ${\mathrm{MN}}^2$t/(M+$N^2$), where t is the reduced temperature. As a function of τ the scaling function scrP(τ,scrr) is a sequence of δ functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=∞. In the latter case the specific heat per lattice site can be written as &,∞), where α is the bulk specific-heat exponent with the known value of (1/2) and ${\nu }_M$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\nu }_m$ by ${\nu }_N$ which takes the value ${\nu }_N$=(1/2). According to finite-size-scaling theory ${\nu }_M$ and ${\nu }_N$ may be interpreted to be the critical exponents ${\nu }_{\mathrm{y}}$ and ${\nu }_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\nu }_m$ and ${\nu }_N$ are in agreement with the values of ${\nu }_{\mathrm{y}}$ and ${\nu }_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.