From Aztec diamonds to pyramids: steep tilings

Abstract

We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of ${\mathbb{Z}}^2$ of the form $1\le x-y\le 2\ell$ for some integer $\ell \ge 1$, and are parametrized by a binary word $w\in \{+,-{\}}^{2\ell }$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $w=(+-{)}^{\ell }$ and to the limit case $w={+}^{\infty }{-}^{\infty }$. For each word $w$ and for different types of boundary conditions, we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips, that admits a natural multivariate generalization. The main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism (which we slightly extend in order to handle Littlewood-type identities). In probabilistic terms our tilings map to Schur processes of different types (standard, Pfaffian and periodic). We also introduce a more general model that interpolates between domino tilings and plane partitions.

36 pages, 22 figures (v3: final accepted version with new Figure 6, new improved proof of Proposition 11)

References in corpus (3)

• Non-commutative Donaldson-Thomas theory and the conifold
• Dimers on Rail Yard Graphs
• Arctic curves of the octahedron equation

Cited by in corpus (5)

• The periodic Schur process and free fermions at finite temperature
• The free boundary Schur process and applications I
• Fourier transform on high-dimensional unitary groups with applications to random tilings
• On the domino shuffle and matrix refactorizations
• Correlations in totally symmetric self-complementary plane partitions