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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 ofrhombi. An interaction disfavors pairs of adjacent rhombi of the same type.This is shown to be a special case of a model of fully-packed loops withinteractions between monomers at distance two along a loop. We solve the lattermodel using Coulomb gas techniques and show that its critical exponents varycontinuously with the interaction strenght. At low temperature it undergoes aKosterlitz-Thouless transition to an ordered phase, which is predicted fromnumerics to occur at a temperature T \sim 110K in the experiments.

The semiflexible fully-packed loop model and interacting rhombus tilings