Laurent skew orthogonal polynomials and related symplectic matrices

Particular class of skew orthogonal polynomials are introduced and investigated, which possess Laurent symmetry. They are also shown to appear as eigenfunctions of symplectic generalized eigenvalue problems. The modification of these polynomials gives some symplectic eigenvalue problem and the corresponding matrix is shown to be equivalent to butterfly matrix, which is a canonical form of symplectic matrices.Comment: 19page

Scaling analysis of stationary probability distributions of random walks on one-dimensional lattices with aperiodic disorder

Stationary probability distributions of one-dimensional random walks on lattices with aperiodic disorder are investigated. The pattern of the distribution is closely related to the diffusional behavior, which depends on the wandering exponent $\Omega$ of the background aperiodic sequence: If $\Omega<0$, the diffusion is normal and the distribution is extended. If $\Omega>0$, the diffusion is ultraslow and the distribution is localized. If $\Omega=0$, the diffusion is anomalous and the distribution is singular, which shows its complex and hierarchical structure. Multifractal analysis are performed in order to characterize these distributions. Extended, localized, and singular distributions are clearly distinguished only by the finite-size scaling behavior of $\alpha_{\rm min}$ and $f(\alpha_{\rm min})$. The multifractal spectrum of the singular distribution agrees well with that of a simple partitioning process.Comment: 21 pages, 10 figure

Finite current stationary states of random walks on one-dimensional lattices with aperiodic disorder

Stationary states of random walks with finite induced drift velocity on one-dimensional lattices with aperiodic disorder are investigated by scaling analysis. Three aperiodic sequences, the Thue-Morse (TM), the paperfolding (PF), and the Rudin-Shapiro (RS) sequences, are used to construct the aperiodic disorder. These are binary sequences, composed of two symbols A and B, and the ratio of the number of As to that of Bs converges to unity in the infinite sequence length limit, but their effects on diffusional behavior are different. For the TM model, the stationary distribution is extended, as in the case without current, and the drift velocity is independent of the system size. For the PF model and the RS model, as the system size increases, the hierarchical and fractal structure and the localized structure, respectively, are broken by a finite current and changed to an extended distribution if the system size becomes larger than a certain threshold value. Correspondingly, the drift velocity is saturated in a large system while in a small system it decreases as the system size increases.Comment: 16 pages, 13 figure

Growth rate distribution of NH_4Cl dendrite and its scaling structure

Scaling structure of the growth rate distribution on the interface of a dendritic pattern is investigated. The distribution is evaluated for an ${\rm NH_4Cl}$ quasi-two-dimensional crystal by numerically solving the Laplace equation with the boundary condition taking account of the surface tension effect. It is found that the distribution has multifractality and the surface tension effect is almost ineffective in the unscreened large growth region. The values of the minimum singular exponent and the fractal dimension are smaller than those for the diffusion-limited aggregation pattern. The Makarov's theorem, the information dimension equals one, and the Turkevich-Scher conjecture between the fractal dimension and the minimum singularity exponent hold.Comment: 5 pages, 6 figure

Multifractal Distribution of Dendrite on One-dimensional Support

We apply multifractal analysis to an experimentally obtained quasi-two-dimensional crystal with fourfold symmetry, in order to characterize the sidebranch structure of a dendritic pattern. In our analysis, the stem of the dendritic pattern is regarded as a one-dimensional support on which a measure is defined and the measure is identified with the area, perimeter length, and growth rate distributions. It is found that these distributions have multifractality and the results for the area and perimeter length distributions, in the competitive growth regime of sidebranches, are phenomenologically understood as a simple partitioning process.Comment: 17 pages, 19 figure

Iterative conformal mapping approach to diffusion-limited aggregation with surface tension effect

We present a simple method for incorporating the surface tension effect into an iterative conformal mapping model of two-dimensional diffusion-limited aggregation. A curvature-dependent growth probability is introduced and the curvature is given by utilizing the branch points of a conformal map. The resulting cluster exhibits a crossover from compact to fractal growth. In the fractal growth regime, it is confirmed, by the conformal map technique, that the fractal dimension of its area and perimeter length coincide.Comment: 18 pages, 10 figure

Quantum Walks on Graphs of the Ordered Hamming Scheme and Spin Networks

It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.Comment: 12 pages, 4 figures, Submission to SciPos

Spin Chains, Graphs and State Revival

Connections between the 1-excitation dynamics of spin lattices and quantum walks on graphs will be surveyed. Attention will be paid to perfect state transfer (PST) and fractional revival (FR) as well as to the role played by orthogonal polynomials in the study of these phenomena. Included is a discussion of the ordered Hamming scheme, its relation to multivariate Krawtchouk polynomials of the Tratnik type, the exploration of quantum walks on graphs of this association scheme and their projection to spin lattices with PST and FR.Comment: 20 pages, based on the lecture delivered by Luc Vinet at the AIMS-Volkwagen workshop 2018 in Douala, Camerou

Pump process of the rotatory molecular motor and its energy efficiency

The pump process of the ratchet model inspired by the $F_o$ rotatory motor of ATP synthase is investigated. In this model there are two kinds of characteristic time. One is dynamical, the relaxation time of the system. Others are chemical, the chemical reaction rates at which a proton binds to or dissociates from the motor protein. The inequalities between them affect the behavior of the physical quantities, such as the rotation velocity and the proton pumping rates across the membrane. The energy transduction efficiency is calculated and the condition under which the efficiency can become higher is discussed. The proton pumping rate and the efficiency have a peak where a certain set of inequalities between the chemical reaction rates and the reciprocal of the relaxation time holds. The efficiency also has a peak for a certain value of the load. The best efficiency condition for the pump process is consistent with that for the motor process.Comment: 8 pages, 13 figure

Quantum State Transfer in a Two-dimensional Regular Spin Lattice of Triangular Shape

Quantum state transfer in a triangular domain of a two-dimensional, equally-spaced, spin lat- tice with non-homogeneous nearest-neighbor couplings is analyzed. An exact solution of the one- excitation dynamics is provided in terms of 2-variable Krawtchouk orthogonal polynomials that have been recently defined. The probability amplitude for an excitation to transit from one site to another is given. For some values of the parameters, perfect transfer is shown to take place from the apex of the lattice to the boundary hypotenuse.Comment: 4 pages, 1 figures; PACS numbers: 03.67.Hk, 02.30.Zz, 02.30.G