Asymptotics of block Toeplitz determinants and the classical dimer model

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We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice (t=0) and the triangular lattice (t=1), and we obtain the asymptotics for 0<t≤ 1. For 0<t<1 we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.