On $\lambda$-determinants and tiling problems

TL;DR

This paper explores the relationship between $\lambda$-determinants, tiling problems, and combinatorial interpretations, particularly focusing on domino tilings of Aztec diamonds and their connection to matrix minors.

Contribution

It provides a combinatorial interpretation of $\lambda$-determinants in terms of domino tilings and reinterprets the Robbins-Rumsey formula as a tiling partition function.

Findings

$\lambda$-determinants correspond to domino tilings of Aztec diamonds.

The Robbins-Rumsey formula is expressed as a tiling partition function.

Connections between recurrence relations and tiling enumerations are established.

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.