  We show that critical exponents of the transition to columnar order in a {\em
mixture} of $2 \times 1$ dimers and $2 \times 2$ hard-squares on the square
lattice {\em depends on the composition of the mixture} in exactly the manner
predicted by the theory of Ashkin-Teller criticality, including in the
hard-square limit. This result settles the question regarding the nature of the
transition in the hard-square lattice gas. It also provides the first example
of a polydisperse system whose critical properties depend on composition. Our
ideas also lead to some interesting predictions for a class of frustrated
quantum magnets that exhibit columnar ordering of the bond-energies at low
temperature.
