Colored vertex models and $k$-tilings of the Aztec diamond

TL;DR

This paper explores weighted $k$-tilings of the Aztec diamond using colored vertex models, deriving generating functions, combinatorial bijections, and arctic curves, extending understanding of tiling configurations and phase boundaries.

Contribution

It introduces a novel approach using colored vertex models to analyze $k$-tilings, providing explicit generating functions, bijections, and arctic curve computations.

Findings

Derived generating polynomials for $k$-tilings.

Established a bijection between interaction-free $k$-tilings and $1$-tilings.

Computed arctic curves for different parameter regimes.

Abstract

We study $k$-tilings ($k$-tuples of domino tilings) of the Aztec diamond of rank $m$. We assign a weight to each $k$-tiling, depending on the number of dominos of certain types and the number of "interactions" between the tilings. Employing the colored vertex models introduced in earlier work to study supersymmetric LLT polynomials, we compute the generating polynomials of the $k$-tilings. We then prove some combinatorial results about $k$-tilings, including a bijection between $k$-tilings with no interactions and $1$-tilings, and we compute the arctic curves of the tilings for $t=0$ and $t\rightarrow\infty$. We also present some lozenge $k$-tilings of the hexagon and compute the arctic curves of the tilings for $t=0$.