Correlated Markov Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on ${\mathbb {Z}^d}$ performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in ${\mathbb {Z}^d}$ for random updates of the coin states of the following form. The random sequences of unitary updates are given by a site-dependent function of a Markov chain in time, with the following properties: on each site, they share the same stationary Markovian distribution and, for each fixed time, they form a deterministic periodic pattern on the lattice. We prove a Feynman–Kac formula to express the characteristic function of the averaged distribution over the randomness at time n in terms of the nth power of an operator M. By analyzing the spectrum of M, we show that this distribution possesses a drift proportional to the time and its centered counterpart displays a diffusive behavior with a diffusion matrix we compute. Moderate and large deviation principles are also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. An example of random updates for which the analysis of the distribution can be performed without averaging is worked out. The random distribution displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix, the law of which we compute. We complete the picture by presenting an uncorrelated example.     

N叉树上的离散时间量子随机行走

通过离散时间量子随机行走的框架,我们研究了在N叉树上的离散时间量子随机行走,该框架不需要硬币空间,仅仅只需要选择一个除了酉性再无其它限制的演化算子,并且包含了使用再生结构的轨道枚举和z变换.作为结果,我们在封闭形式中计算了在根处的振幅的生成函数.     

Discrete Quantum Scattering Theory

We formulate quantum scattering theory in terms of a discrete L ₂-basis of eigen differentials. Using projection operators in the Hilbert space, we develop a universal method for constructing finite-dimensional analogues of the basic operators of the scattering theory: S- and T-matrices, resolvent operators, and Möller wave operators as well as the analogues of resolvent identities and the Lippmann–Schwinger equations for the T-matrix. The developed general formalism of the discrete scattering theory results in a very simple calculation scheme for a broad class of interaction operators.     

Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments

We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Lévy process with heavy-tailed Lévy measure. A central point of the paper is that we make full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Lévy stochastic networks.     

Eigenvalues of Quantum Walks of Grover and Fourier Types

A necessary and sufficient conditions for a certain class of periodic unitary transition operators to have eigenvalues are given. Applying this, it is shown that Grover walks in any dimension has both of \(\pm \, 1\) as eigenvalues and it has no other eigenvalues. It is also shown that the lazy Grover walks in any dimension has 1 as an eigenvalue, and it has no other eigenvalues. As a result, a localization phenomenon occurs for these quantum walks. A general conditions for the existence of eigenvalues can be applied also to certain quantum walks of Fourier type. It is shown that the two-dimensional Fourier walk does not have eigenvalues and hence it is not localized at any point. Some other topics, such as Grover walks on the triangular lattice, products and deformations of Grover walks, are also discussed.

     

Compactifications of Discrete Quantum Groups

Given a discrete quantum group we construct a Hopf -algebra which is a unital -subalgebra of the multiplier algebra of . The structure maps for are inherited from and thus the construction yields a compactification of which is analogous to the Bohr compactification of a locally compact group. This algebra has the expected universal property with respect to homomorphisms from multiplier Hopf algebras of compact type (and is therefore unique). This provides an easy proof of the fact that for a discrete quantum group with an infinite dimensional algebra the multiplier algebra is never a Hopf algebra.Partially supported by Komitet Badań Naukowych grants 2P03A04022 & 2P03A01324, the Foundation for Polish Science and Deutsche Forschungsgemeinschaft.     

Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of ‘symmetry’ in the spatial configuration of branching sources. The presented results are based on Green’s function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.     

On the Discrete Spectrum of Generalized Quantum Tubes

The spectrum (of the Dirichlet Laplacian) of non-compact, non-complete Riemannian manifolds is much less understood than their compact counterparts. In particular it is often not even known whether such a manifold has any discrete spectra. In this article, we will prove that a certain type of non-compact, non-complete manifold called the quantum tube has non-empty discrete spectrum. The quantum tube is a tubular neighborhood built about an immersed complete manifold in Euclidean space. The terminology of “quantum” implies that the geometry of the underlying complete manifold can induce discrete spectra – hence quantization. We will show how the Weyl tube invariants appear in determining the existence of discrete spectra. This is an extension and generalization, on the geometric side, of the previous work of the author on the “quantum layer.”     

Discrete Valuations Extend to Certain Algebras of Quantum Type

So-called “quantized” algebras are popular objects of study in non-commutative algebra. Usually such algebras are either positively graded Ore domains R with R ₀ = K a field and R = K[R ₁], R ₁ being a finite dimensional K vectorspace, or else filtered rings having a ring of forementioned type for its associated graded ring. We show that every discrete valuation of K extends to a valuation, in the sence of O. Schilling (cf. [S]), of the skewfield of fractions, Δ = Q_cl (R), of the Ore domain R (Proposition 2.3. and Corollary 2.8.). Such extension property has long been known to fail for finite dimensional skewfields over K; its validity in the case of several quantized algebras may be viewed as a consequence of the rigidity of their defining relations. Our result opens the door for a more arithmetical study of Δ e.g. in case K is a numberfield or an algebraic function field of a curve; for an application in this direction we refer to a first version of some divisor calculus started in [VW].     

Coefficient Matrices of a Quantum Discrete Auxiliary Linear Problem

In the present paper, we describe general properties of quantum matrices that are coefficient matrices of an auxiliary problem for quantum discrete three-dimensional integrable models. Our goal is to prove a universal functional equation for the quantum determinant in the case of a finite-dimensional representation of a local Weyl algebra. Bibliography: 4 titles.     

A New Class of Fully Discrete Sparse Fourier Transforms: Faster Stable Implementations with Guarantees

Journal of Fourier Analysis and Applications - In this paper we consider sparse Fourier transform (SFT) algorithms for approximately computing the best s-term approximation of the discrete Fourier...     

Exponentially many supertrees

The amalgamation of leaf-labelled trees into a single supertree that displays each of the input trees is an important problem in classification. Clearly, there can be more than one (super) tree for a given set of input trees, in particular if a highly unresolved supertree exists. Here, we show (by explicit construction) that even if every supertree of a given collection of input trees is binary, there can still be exponentially many such supertrees.     

指数型调和映射

证明了在适当条件下,指数型能量和指数型调和映射是共形不变的.我们主要研究了指数型调和的黎曼淹没和等距浸入,还研究了与黎曼等距浸入相关的高斯映射是指数型调和.     

Exponentially confining potential well

Theoretical and Mathematical Physics - We introduce an exponentially confining potential well that can be used as a model to describe the structure of a strongly localized system. We obtain an...     

Gaussian Decomposition of Quantum Groups and a Model of Discrete Dynamics

A discrete dynamical model based on the Gaussian decomposition of the quantum group GL_q(2) is considered. Bibliography: 10 titles.     

Exponentially Many Hadamard Designs

If there is a Hadamard design of order n, then there are at least 2^{8n−16−9log n} non-isomorphic Hadamard designs of order 2n. Mathematics Subject Classificaion 2000: 05B05     

A Remark on Twists and the Notion of Torsion-free Discrete Quantum Groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki–Takai duality. The twisted Takesaki–Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld–Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.