Using exact methods, pair-correlation functions are studied in the dimer model defined on a brick lattice. At long distances these functions exhibit strongly anisotropic algebraic decay and, near criticality, the length scales diverge differently in the two principal directions. The critical exponents are v_x = 1/2 and v_y = 1. These results are in agreement with deductions drawn from recent exact finite-size scaling calculations. We also interpret our results in the light of domain wall theories of commensurate-incommensurate transitions, and in particular we study the relation of the present model to the discrete version of the Pokrovsky Talapov model introduced by Villain.