Related papers: Square ice, alternating sign matrices and classica…
We study the annealed complexity of a random Gaussian homogeneous polynomial on the $N$-dimensional unit sphere in the presence of deterministic polynomials that depend on fixed unit vectors and external parameters. In particular, we…
We study the Hankel determinant and orthogonal polynomials with respect to the two-parameter weight function $$ w(x)=w(x;t_1, t_2):=\exp(-x^6-t_2 x^4-t_1 x^2),\qquad x\in\mathbb{R}, $$ with $t_1,\; t_2 \in \mathbb{R}$. This problem arises…
We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and…
The nineteen-vertex model on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial…
We study the partition function for the three-colour model with domain wall boundary conditions. We express it in terms of certain special polynomials, which can be constructed recursively. Our method generalizes Kuperberg's proof of the…
We study a one-parameter family ($\ell=1,2,3,\ldots$) of configurations that are square-ice analogues of plane partitions. Using an algorithm due to Bratley and McKay, we carry out exact enumerations in order to study their asymptotic…
In this note, we consider the six-vertex model with domain wall boundary conditions, defined on a $M\times M$ lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities $\lambda_j$ and $\mu_k$. For a…
We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends…
This work explores classical discrete multiple orthogonal polynomials, including Hahn, Meixner of the first and second kinds, Kravchuk, and Charlier polynomials, with an arbitrary number of weights. Explicit expressions for the recursion…
One deals with degenerations by coordinate sections of the square generic Hankel matrix over a field $k$ of characteristic zero, along with its main related structures, such as the determinant of the matrix, the ideal generated by its…
This article focuses on estimating distribution elements over a high-dimensional binary hypercube from multivariate binary data. A popular approach to this problem, optimizing Walsh basis coefficients, is made more interpretable by an…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-\alpha)(x-\beta)\right]^{-\frac{1}{2}}$, $x\in[0,\alpha]$, $0<\alpha<\beta$. A related system was studied by C. J. Rees in 1945,…
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit…
This paper is the second in a series of planned papers which provide first bijective proofs of alternating sign matrix results. Based on the main result from the first paper, we construct a bijective proof of the enumeration formula for…
This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schr\"{o}dinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing…
Boundary conditions compatible with integrability are obtained for two dimensional models by solving the factorizability equations for the reflection matrices $K^{\pm}(\theta)$. For the six vertex model the general solution depending on…
There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is an open problem for about $40$ years now. So…
We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified…
Halved monotone triangles are a generalisation of vertically symmetric alternating sign matrices (VSASMs). We provide a weighted enumeration of halved monotone triangles with respect to a parameter which generalises the number of $-1$s in a…
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free…