On λ-determinants and tiling problems

Abstract

We review the connections between the octahedral recurrence, λ-determinants and tiling problems. This provides in particular a direct combinatorial interpretation of the λ-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.