On $\lambda$-determinants and tiling problems

Abstract

We review the connections between the octahedral recurrence, $\lambda$-determinants and tiling problems. This provides in particular a direct combinatorial interpretation of the $\lambda$-determinant (and generalizations thereof) of an arbitrary matrix in terms of domino tilings of Aztec diamonds. We also reinterpret the general Robbins-Rumsey formula for the rational function of consecutive minors, given by a summation over pairs of compatible alternating sign matrices, as the partition function for tilings of Aztec diamonds equipped with a general measure.