  We consider a model of weakly interacting, close-packed, dimers on the
two-dimensional square lattice. In a previous paper, we computed both the
multipoint dimer correlations, which display non-trivial critical exponents,
continuously varying with the interaction strength; and the height
fluctuations, which, after proper coarse graining and rescaling, converge to
the massless Gaussian field with a suitable interaction-dependent pre-factor
(`amplitude'). In this paper, we prove the identity between the critical
exponent of the two-point dimer correlation and the amplitude of this massless
Gaussian field. This identity is the restatement, in the context of interacting
dimers, of one of the Haldane universality relations, part of his Luttinger
liquid conjecture, originally formulated in the context of one-dimensional
interacting Fermi systems. Its validity is a strong confirmation of the
effective massless Gaussian field description of the interacting dimer model,
which was guessed on the basis of formal bosonization arguments. We also
conjecture that a certain discrete curve defined at the lattice level via the
Temperley bijection converges in the scaling limit to an SLE$_\kappa$ process,
with $\kappa$ depending non-trivially on the interaction and related in a
simple way to the amplitude of the limiting Gaussian field.
