From Aztec diamonds to pyramids: Steep tilings - Archive ouverte HAL
Article Dans Une Revue Transactions of the American Mathematical Society Année : 2017

From Aztec diamonds to pyramids: Steep tilings

Résumé

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 $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$, 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.

Dates et versions

hal-01017723 , version 1 (03-07-2014)

Identifiants

Citer

Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel. From Aztec diamonds to pyramids: Steep tilings. Transactions of the American Mathematical Society, 2017, 369 (8), pp.5921 - 5959. ⟨10.1090/tran/7169⟩. ⟨hal-01017723⟩
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