Motivated by a recent adsorption experiment [M.O. Blunt et al., Science 322, 1077 (2008)], we study tilings of the plane with three different types of rhombi. An interaction disfavors pairs of adjacent rhombi of the same type. This is shown to be a s
pecial case of a model of fully-packed loops with interactions between monomers at distance two along a loop. We solve the latter model using Coulomb gas techniques and show that its critical exponents vary continuously with the interaction strenght. At low temperature it undergoes a Kosterlitz-Thouless transition to an ordered phase, which is predicted from numerics to occur at a temperature T sim 110K in the experiments.
