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We study the asymptotics of bounded lecture hall tableaux. Limit shapes form when the bounds of the lecture hall tableaux go to infinity linearly in the lengths of the partitions describing the large-scale shapes of these tableaux. We prove Conjecture 6.1 in \cite{SKN21}, stating that the slopes of the rescaled height functions in the scaling limit satisfy a complex Burgers equation. We also show that the fluctuations of the unrescaled height functions converge to the Gaussian free field. The proof is based on new construction and analysis of Schur generating functions for the lecture hall tableaux, whose corresponding particle configurations do not form a Gelfand-Tsetlin scheme; and the corresponding dimer models are not doubly periodic.

In this article we define a generalization of the domino shuffling algorithm for tilings of the Aztec diamond to the interacting $k$-tilings recently introduced by S. Corteel, A. Gitlin, and the first author. We describe the algorithm both in terms of dynamics on a system of colored particles and as operations on the dominos themselves., Comment: 43 pages, 18 figures

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 $t=0$., Comment: 41 pages

We describe a Yang-Baxter integrable vertex model, which can be realized as a degeneration of a vertex model introduced by Aggarwal, Borodin, and Wheeler. From this vertex model, we construct a certain class of partition functions that we show are essentially equal to the super ribbon functions of Lam. Using the vertex model formalism, we give proofs of many properties of these polynomials, namely a Cauchy identity and generalizations of known identities for supersymmetric Schur polynomials.