Matrix biorthogonal polynomials on the unit circle and non-Abelian Ablowitz-Ladik hierarchy

Adler and van Moerbeke \cite{AVM} described a reduction of 2D-Toda hierarchy called Toeplitz lattice. This hierarchy turns out to be equivalent to the one originally described by Ablowitz and Ladik \cite{AL} using semidiscrete zero-curvature equations. In this paper we obtain the original semidiscrete zero-curvature equations starting directly from the Toeplitz lattice and we generalize these computations to the matrix case. This generalization lead us to the semidiscrete zero-curvature equations for the non-abelian (or multicomponent) version of Ablowitz-Ladik equations \cite{GI}. In this way we extend the link between biorthogonal polynomials on the unit circle and Ablowitz-Ladik hierarchy to the matrix case.Comment: 23 pages, accepted on publication on J. Phys. A., electronic link: http://stacks.iop.org/1751-8121/42/36521

On the Distributions of the Lengths of the Longest Monotone Subsequences in Random Words

We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlev\'e V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k x k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N -> infinity limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k x k hermitian matrices of trace zero.Comment: 30 pages, revised version corrects an error in the statement of Theorem

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

This paper introduces and analyses the new grid-based tensor approach to approximate solution of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree-Fock equation over a spatial $L_1\times L_2\times L_3$ lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large $L_1\times L_2\times L_3$ lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic $L\times 1\times 1$ lattice chain in a 3D rectangular "tube" with $L$ up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large $L$.Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with arXiv:1408.383

High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets

We study monic polynomials $Q_n(x)$ generated by a high order three-term recursion $xQ_n(x)=Q_{n+1}(x)+a_{n-p} Q_{n-p}(x)$ with arbitrary $p\geq 1$ and $a_n>0$ for all $n$. The recursion is encoded by a two-diagonal Hessenberg operator $H$. One of our main results is that, for periodic coefficients $a_n$ and under certain conditions, the $Q_n$ are multiple orthogonal polynomials with respect to a Nikishin system of orthogonality measures supported on star-like sets in the complex plane. This improves a recent result of Aptekarev-Kalyagin-Saff where a formal connection with Nikishin systems was obtained in the case when $\sum_{n=0}^{\infty}|a_n-a|0$. An important tool in this paper is the study of "Riemann-Hilbert minors", or equivalently, the "generalized eigenvalues" of the Hessenberg matrix $H$. We prove interlacing relations for the generalized eigenvalues by using totally positive matrices. In the case of asymptotically periodic coefficients $a_n$, we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain a connection with a vector equilibrium problem. We anticipate that in the future, the study of Riemann-Hilbert minors may prove useful for more general classes of multiple orthogonal polynomials.Comment: 59 pages, 3 figure

Inverse moment problem for elementary co-adjoint orbits

We give a solution to the inverse moment problem for a certain class of Hessenberg and symmetric matrices related to integrable lattices of Toda type.Comment: 13 page