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David Keating

Publications and source records attributed to David Keating.

17 recordsLinked to original sources
arXiv2025

Limit shapes for Domain-Wall (colored) vertex models

We study partition functions with domain-wall like boundary conditions for path models issued from colored vertex models. These models display an arctic phenomenon, as attested by numerical simulations. We show that the colored vertex model is equivalent to a certain single-color ``colorblind" vertex model. In a special case of the weights for the colorblind touching paths, we derive the arctic curve using a bijective sliding map to non-intersecting paths, for which arctic curves were previously derived using the tangent method. The resulting arctic curves are only piecewise analytic, as in the known non-free fermion cases of Six vertex model with domain-wall boundaries and its relatives. We also prove a shear phenomenon, that some portions of the arctic curve are sheared versions of the analytic continuation of other portions, as already observed in the uniformly weighted Six and Twenty vertex models.

math-ph↗
arXiv2025

Diffusive Scaling limit of stochastic Box-Ball systems and PushTASEP

We introduce the Stochastic Box-Ball System (SBBS), a probabilistic cellular automaton that generalizes the classic Takahashi-Satsuma Box-Ball System. In SBBS, particles are transported by a carrier with a fixed capacity that may fail to pick up any given particle with a fixed probability $\epsilon$. This model interpolates between two known integrable systems: the Box-Ball System (as $\epsilon\rightarrow 0$) and the PushTASEP (as $\epsilon\rightarrow 1$). We show that the long-term behavior of SBBS is governed by isolated particles and the occasional emergence of short solitons, which can form longer solitons but are more likely to fall apart. More precisely, we first show that all particles are isolated except for a $1/\sqrt{n}$-fraction of times in any given $n$ steps, and solitons keep forming for this fraction of times. We then show that under diffusive scaling, both SBBS (for any carrier capacity) and PushTASEP converge weakly to semimartingale reflecting Brownian Motions (SRBMs) on the Weyl chamber with explicit covariance and reflection matrices, which are consistent with the microscale relations between these systems. The reflection matrix for SBBS is determined by how 2-solitons behave and exhibit ``solitonic bias'' visible in the diffusive scale. Our proof relies on a new, extended SRBM invariance principle that we develop in this work. This principle can handle processes with complex boundary behavior that can be written as "overdetermined" Skorokhod decompositions, which is crucial for analyzing the complex solitonic interaction in SBBS. We believe this tool may be of independent interest.

math.PR↗
arXiv2025

Perfect t-Embeddings of Uniformly Weighted Generalized Tower Graphs

In this paper, we study sequences of perfect t-embeddings of a uniformly weighted family of graphs we call generalized tower graphs. We show that the embeddings of these graphs satisfy certain technical assumptions, in particular, the rigidity assumption of Berggren-Nicoletti-Russkikh. As a result, we confirm the convergence of the gradients of the height function fluctuations of these graphs to those of the Gaussian free field.

math-ph↗
arXiv2025

Colored Vertex Models and Interacting Reverse Plane Partitions

We study the coupling of pairs of reverse plane partitions of the same shape by assigning a certain local interaction between the reverse plane partitions. We show that they are in bijection with a certain Yang-Baxter integrable colored vertex model. By utilizing the Yang-Baxter equation for this colored vertex model, we are able to compute the generating function for the interacting pairs of reverse plane partitions. We also give a bijection between the coupled pairs of reverse plane partitions with the interaction strength set to zero and a single reverse plane partition of the same shape.

math.CO↗
arXiv2024

Airy limit for $\beta$-additions through Dunkl operators

It is well known that the edge limit of Gaussian/Laguerre Beta-ensembles, as well as a large class of $\beta$-ensembles is given by the $\mathrm{Airy}(\beta)$ point process. We extend this universality result to a general class of additions of Gaussian and Laguerre ensembles, which were identified in \cite{AN} as projection of the ergodic measures of the $\beta$-corners process. In order to make sense of the $\beta$-addition, we introduce the Type-A Bessel function as the characteristic function of our matrix ensemble, following the approach of \cite{GM}, \cite{BCG}. Then we extract its moment information through the action of Dunkl operators, and obtain certain limiting functional expressed via conditional Brownian bridges for the Laplace transform of $\mathrm{Airy}(\beta)$. Our limit expression is universal up to proper rescaling among all of our additions, and agrees with the single-time Laplace transform expression from the concurrent work \cite{GXZ}.

math.PR↗
arXiv2023

Asymptotics of Bounded Lecture-Hall Tableaux

We study weighted bounded \(n\)-lecture hall tableaux with bound \(t_n\) in the joint limit \(n,t_n\to\infty\) with \(t_n/n\to\alpha\in(0,\infty)\). Using their non-intersecting-path representation, we construct a Schur generating function adapted to this model and derive moment formulas for the limiting counting measures. When the normalized tail sums of the weights have a \(C^2\) macroscopic profile with negative derivative, the rescaled height functions converge in probability in \(L^1_{\mathrm{loc}}(\mathbb R\times(0,\alpha))\) to a deterministic limit shape whose complex slope satisfies a Burgers equation with an explicit source term. In the uniformly weighted model of \cite{SKN21}, the source term vanishes, proving Conjecture~6.1 of that work. Under an additional interval-structure assumption on the limiting bottom boundary, and only continuity and strict monotonicity of the tail profile, the unrescaled height fluctuations converge to the Gaussian free field in the sense of joint moments of horizontal polynomial observables. The method applies although the particle configurations are not Gelfand--Tsetlin patterns and the associated dimer model is not doubly periodic.

math.PR↗
arXiv2023

Shuffling algorithm for coupled tilings of the Aztec diamond

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.

math.CO↗
arXiv2022

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

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 $t=0$.

math.CO↗
arXiv2021

A Vertex Model for Supersymmetric LLT Polynomials

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.

math.CO↗
arXiv2021

Equivalences of LLT polynomials via lattice paths

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 certain vertex model whose boundary conditions are determined by $\mathbf{\beta}/\mathbf{\gamma}$. In this paper we describe an algorithm which gives a bijection between the configurations of the vertex model with boundary condition $\mathbf{\beta}/\mathbf{\gamma} = (\beta^{(1)}/\gamma^{(1)},\beta^{(2)}/\gamma^{(2)})$ and those with boundary condition $(\mathbf{\beta}/\mathbf{\gamma})_{swap} = (\beta^{(2)}/\gamma^{(2)},\beta^{(1)}/\gamma^{(1)})$. We prove a sufficient condition for when this bijection is weight-preserving up to an overall factor of $t$, which in turn implies that the corresponding LLT polynomials are equal up to the same overall factor. Using these techniques, we are also able to systematically determine linear relations within families of LLT polynomials.

math.CO↗
arXiv2020

A vertex model for LLT polynomials

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 identity.

math.CO↗
arXiv2020

Area Statistics for Large Oscillating Tableaux

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 converges to a Gaussian stochastic process.

math.CO↗
arXiv2019

Arctic curves phenomena for bounded lecture hall Tableaux

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 tableaux thanks to two other combinatorial models: non intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octogons. We use the tangent method to investigate the arctic curve in the model of nonintersecting lattice paths with fixed starting points and ending points distributibuted according to some arbitrary piecewise differentiable function. We then study the dimer model and use some ansatz to guess the asymptotics of the inverse of the Kasteleyn matrix confirm the arctic curve computed with the tangent method for two examples.

math.CO↗
arXiv2018

Random Tilings with the GPU

We present GPU accelerated implementations of Markov chain algorithms to sample random tilings, dimers, and the six-vertex model.

cs.OH↗
arXiv2017

Validation and benchmarking of two particle-in-cell codes for a glow discharge

The two particle-in-cell codes EDIPIC and LSP are benchmarked and validated for a parallel-plate glow discharge in helium, in which the axial electric field had been carefully measured, primarily to investigate and improve the fidelity of their collision models. The scattering anisotropy of electron-impact ionization, as well as the value of the secondary-electron emission yield, are not well known in this case. The experimental uncertainty for the emission yield corresponds to a factor of two variation in the cathode current. If the emission yield is tuned to make the cathode current computed by each code match the experiment, the computed electric fields are in excellent agreement with each other, and within about 10\% of the experimental value. The non-monotonic variation of the width of the cathode fall with the applied voltage seen in the experiment is reproduced by both codes. The electron temperature in the negative glow is within experimental error bars for both codes, but the density of slow trapped electrons is underestimated. A more detailed code comparison done for several synthetic cases of electron-beam injection into helium gas shows that the codes are in excellent agreement for ionization rate, as well as for elastic and excitation collisions with isotropic scattering pattern. The remaining significant discrepancies between the two codes are due to differences in their electron binary-collision models, and for anisotropic scattering due to elastic and excitation collisions.

physics.plasm-ph↗
arXiv2014

Deterministic Domain Wall Motion Orthogonal To Current Flow Due To Spin Orbit Torque

Deterministic control of domain walls orthogonal to the direction of current flow is demonstrated by exploiting spin orbit torque in a perpendicularly polarized Ta/CoFeB/MgO multilayer in presence of an in-plane magnetic field. Notably, such orthogonal motion with respect to current flow is not possible from traditional spin transfer torque driven domain wall propagation even in presence of an external magnetic field. Reversing the polarity of either the current flow or the in-plane field is found to reverse the direction of the domain wall motion. From these measurements, which are unaffected by any conventional spin transfer torque by symmetry, we estimate the spin orbit torque efficiency of Ta to be 0.08.

cond-mat.mtrl-sci↗