David Keating's research while affiliated with University of California and other places

Publications (12)

Article
Full-text available
In this paper we prove that the Euler–Lagrange equations for the limit shape for the inhomogeneous six vertex model on a cylinder have infinitely many conserved quantities.
Preprint
Full-text available
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...
Preprint
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 proo...
Article
We describe a novel Yang–Baxter integrable vertex model. From this vertex model we construct a certain class of partition functions that we show are essentially equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we give alternate proofs of many properties of these polynomials, including symmetry and a Ca...
Preprint
The LLT polynomials $\mathcal{L}_{\mathbf{\beta}/\mathbf{\gamma}} (X;t)$ are a family of symmetric polynomials indexed by a tuple of (possibly skew-)partitions $\mathbf{\beta}/\mathbf{\gamma}= (\beta^{(1)}/\gamma^{(1)},\ldots,\beta^{(k)}/\gamma^{(k)})$. It has recently been shown that these polynomials can be seen as the partition function of a cer...
Article
Full-text available
Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded t...
Preprint
We describe a novel Yang-Baxter integrable vertex model. From this vertex model we construct a certain class of partition functions that we show are equal to the LLT polynomials of Lascoux, Leclerc, and Thibon. Using the vertex model formalism, we give alternate proofs of many properties of these polynomials, including symmetry and a Cauchy identit...
Preprint
In this note we show that the area of the partitions making up an oscillating tableaux is described by a random walk on the first quadrant of $\mathbb{Z}^2$ with certain position dependent weights. We are able to recursively calculate the moments of the walk. As the length of the oscillating tableaux becomes large we show that this random walk conv...
Preprint
In this paper we prove that the Euler-Lagrange equations for the limit shape for the inhomogeneous six vertex model on a cylinder have infinitely many conserved quantities.
Article
Full-text available
In this note, we derive the asymptotical behavior of local correlation functions in dimer models on a domain of the hexagonal lattice in the continuum limit, when the size of the domain goes to infinity and the parameters of the model scale appropriately.
Preprint
Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded t...
Article
We present GPU accelerated implementations of Markov chain algorithms to sample random tilings, dimers, and the six-vertex model.

Citations

... A prototypical example is provided by the six-vertex model with domain wall boundary conditions. Current interest in the model is mostly motivated by the occurrence of phase separation [2][3][4][5], which recently triggered a number of numerical studies [6][7][8] and analytical results [9][10][11][12]. The model is also of relevance for quantum quenches in the closely related Heisenberg XXZ quantum spin chain [13][14][15][16], and for N = 4 super-Yang-Mills theory [17][18][19]. ...
... The main goal of this paper is to study superpositions of domino tilings of the Aztec diamond, by using recent ideas coming from the study of colored vertex models. It has recently been shown [1,6,7] that the LLT polynomials can be realized as a certain class of partition functions constructed from an integrable vertex model, and more recently the same was shown for the supersymmetric LLT polynomials (also called super ribbon functions) [8,12]. We will use the vertex models introduced in [12], which can also be realized as degenerations of a vertex model introduced in [1]. ...
... In the thermodynamic limit, obtained by sending the mesh size to 0 while keeping the size of the domain unchanged, this separation is the so-called arctic curve of the model. Without being exhaustive, such an arctic phenomenon is observed in lozenge tilings of hexagons [1], in domino tilings of Aztec rectangles with defects [2], in bounded lecture hall tableaux [3] or in configuations of the six-vertex (6V) with various boundary conditions [4]. ...
... (In fact Fig. 31 displays a neighborhood of (x, y) with a slightly different embedding of the graph from what we have defined above. The computations that follow are unaffected by this up to first order, so we omit details. ) We guess the form of the inverse of the Kasteleyn matrix using an ansatz developed by Keating, Reshetikhin and Sridhar [21]. They developed such an ansatz for the dimer model on the hexagonal lattice. ...
... One such tiling is shown in Figure 10. We used the source code developed by Keating and Sridhar [KS18a] described in [KS18b], with some modifications. We ran our code on a GTX 1080 Ti GPU. ...