Search

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.

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.

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.

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.

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.

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., Comment: 45 pages

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

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.

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., Comment: 10 pages of main manuscript, 4 figures, 20 pages of Supplementary Information