Lattice Zd Research Articles

On the Nadaraya–Watson kernel regression estimator for irregularly spaced spatial data

We investigate the asymptotic normality of the Nadaraya–Watson kernel regression estimator for irregularly spaced data collected on a finite region of the lattice Zd where d is a positive integer. The results are stated for strongly mixing random fields in the sense of Rosenblatt (1956) and for weakly dependent random fields in the sense of Wu (2005). Only minimal conditions on the bandwidth parameter and simple conditions on the dependence structure of the data are assumed.

Algebraic spectral synthesis and crystal rigidity

A spectral synthesis property is obtained for closed shift-invariant subspaces of vector-valued functions on the lattice Zd. This result generalises Marcel Lefranc's 1958 theorem for scalar-valued functions. Applications are given to homogeneous systems of multivariable vector-valued discrete difference equations and to the first-order flexibility of crystallographic bar-joint frameworks.

The critical infection rate of the high-dimensional two-stage contact process

In this paper we are concerned with the two-stage contact process on the lattice Zd introduced in Krone (1999). We give a limit theorem of the critical infection rate of the process as the dimension d of the lattice grows to infinity. A linear system and a two-stage SIR model are two main tools for the proof of our main result.

Bounds on the discrete spectrum of lattice Schrödinger operators

We discuss the validity of the Weyl asymptotics—in the sense of two-sided bounds—for the size of the discrete spectrum of (discrete) Schrödinger operators on the d-dimensional, d ≥ 1, cubic lattice Zd at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension d ≥ 1—even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions d ≥ 1 that, for potentials well behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case d ≥ 3 while stronger for d = 1, 2. It is well known that the semi-classical number of bound states is—up to a constant—always an upper bound on the size of the discrete spectrum of Schrödinger operators if d ≥ 3. We show here how to construct general upper bounds on the number of bound states of Schrödinger operators on Zd from semi-classical quantities in all space dimensions d ≥ 1 and independently of the positivity-improving property of the free Hamiltonian.

The essential spectrum of the three-particle discrete operator corresponding to a system of three fermions on a lattice

We consider a family of three-particle discrete Shrodinger operators H μ (K). These operators are associated with the Hamiltonian for a system of three identical particles (fermions) with pairwise two-particle interactions on neighboring junctions of the d-dimensional lattice Z d . We describe the location and the structure of the essential spectrum of the operator H μ (K) for all values of the three-particle quasi-momentum K ∈ T d and the interaction energy μ > 0.

Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Zd at small inverse temperature and dimension d≥3. Then, the normalized partition function Wn is a regular martingale with limit W. We prove that n(d−2)/4(Wn−W)/Wn converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale Wn are different from those for polymers on trees.

Mean field limit for survival probability of the high-dimensional contact process

In this paper we are concerned with the contact process on the hypercube lattice Zd. The contact process intuitively describes the spread of the infectious disease on a graph, where an infectious vertex becomes healthy at a constant rate while a healthy vertex is infected at rate proportional to the number of infectious neighbors. As the dimension of the lattice grows to infinity, we give a mean field limit for the survival probability of the process conditioned on the event that only the origin of the lattice is infected at t=0. The binary contact path process is a main auxiliary tool for our proof.

Exact convergence rate of the local limit theorem for branching random walks on the integer lattice

Consider branching random walks on the integer lattice Zd, where the branching mechanism is governed by a supercritical Galton–Watson process and the particles perform a symmetric nearest-neighbor random walk whose increments equal to zero with probability r∈[0,1). We derive exact convergence rate in the local limit theorem for distributions of particles. When r=0, our results correct and improve the existing results on the convergence speed conjectured by Révész (1994) and proved by Chen (2001). As a byproduct, we obtain exact convergence rate in the local limit theorem for some symmetric nearest-neighbor random walks, which is of independent interest.

The Range of a Rotor Walk

In rotor walk on a graph, the exits from each vertex follow a prescribed periodic sequence. We show that any rotor walk on the d-dimensional lattice ℤd visits at least on the order of td/(d+1) distinct sites in t steps. This result extends to Eulerian graphs with a volume growth condition. In a uniform rotor walk, the first exit from each vertex is to a neighbor chosen uniformly at random. We prove a shape theorem for the uniform rotor walk on the comb graph, showing that the size of the range is of order t2/3 and the asymptotic shape of the range is a diamond. Using a connection to the mirror model, we show that the uniform rotor walk is recurrent on two different directed graphs obtained by orienting the edges of the square grid: the Manhattan lattice and the F-lattice. We end with a short discussion of the time it takes for rotor walk to cover a finite Eulerian graph.

Gaussian bounds and collisions of variable speed random walks on lattices with power law conductances

We consider a weighted lattice Zd with conductance μe=∣e∣−α. We show that the heat kernel of a variable speed random walk on it satisfies a two-sided Gaussian bound by using an intrinsic metric. We also show that when d=2 and α∈(−1,0), two independent random walks on such weighted lattice will collide infinitely many times while they are transient.

Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd

Real Gibbs measures arise in many problems of probability theory and statistical mechanics. This measure, related to the Boltzmann distribution, generalizes the notion of canonical ensemble. In addition, Gibbs measure is unique measure that maximizes the entropy of the expected energy. But non-archimedean (p-adic) analogue of Gibbs measures have been little studied. It is known that in the case of real numbers concepts of Gibbs measure and Markov random field are identical. But in the p-adic case, the class of p-adic Markov random fields is wider than the class of p-adic Gibbs measures [1]. One of the main problems of physics is to study the set of all p-adic Gibbs measures (see e.g. [1, 2]). Let us present some main definitions from the theory of p-adic numbers (see [3–5]). Let p be a prime number. Every rational number x = 0 can be represented in the form x = p n m , where r, n ∈ Z, m is a positive number, (n,m) = 1, where m and n are not divisible by p. A p-adic norm of rational number x = p n m is defined as follows

Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs

A gain graph is a graph whose edges are orientably labeled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). In order to do that we develop a relative of the Tutte polynomial of a semimatroid. Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found.An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Zd, the lists are order ideals in the integer lattice Zd, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph.This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n×d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.

Inhomogeneous Long-Range Percolation for Real-Life Network Modeling

The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice Zd, d ≥ 1, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.

A Lower Bound for the End-to-End Distance of the Self-Avoiding Walk

AbstractFor an N-step self-avoiding walk on the hypercubic lattice Zd, we prove that the meansquare end-to-end distance is at least N4=(3d) times a constant. This implies that the associated critical exponent v is at least 2/(3d), assuming that v exists.

Positivity of eigenvalues of the two-particle Schrödinger operator on a lattice

We consider the family H(k) of two-particle discrete Schrodinger operators depending on the quasimomentum of a two-particle system k ∈ \(\mathbb{T}^d \), where \(\mathbb{T}^d \) is a d-dimensional torus. This family of operators is associated with the Hamiltonian of a system of two arbitrary particles on the d-dimensional lattice ℤd, d ≥ 3, interacting via a short-range attractive pair potential. We prove that the eigenvalues of the Schrodinger operator H(k) below the essential spectrum are positive for all nonzero values of the quasimomentum k ∈ \(\mathbb{T}^d \) if the operator H(0) is nonnegative. We establish a similar result for the eigenvalues of the Schrodinger operator H+(k), k ∈ \(\mathbb{T}^d \), corresponding to a two-particle system with repulsive interaction.

Harmonic labeling of graphs

Which graphs admit an integer valued harmonic function which is injective and surjective onto Z? Such a function, which we call harmonic labeling, is constructed when the graph is the lattice Zd. It is shown that for any finite graph G containing at least one edge, there is no harmonic labeling of G×Z. Further discussion and open problems are presented.

Catalytic branching random walks in ℤ d with branching at the origin

A time-continuous branching random walk on the lattice ℤd, d ≥ 1, is considered when the particles may produce offspring at the origin only. We assume that the underlying Markov random walk is homogeneous and symmetric, the process is initiated at moment t = 0 by a single particle located at the origin, and the average number of offspring produced at the origin is such that the corresponding branching random walk is critical. The asymptotic behavior of the survival probability of such a process at moment t → ∞ and the presence of at least one particle at the origin is studied. In addition, we obtain the asymptotic expansions for the expectation of the number of particles at the origin and prove Yaglom-type conditional limit theorems for the number of particles located at the origin and beyond at moment t.

Convergence of the Abelian sandpile

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice Zd, in which sites with at least 2d chips topple, distributing one chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of n chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as n→∞. However, little has been proved about the appearance of the stable configurations. We use partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as n→∞. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

Linearly repetitive Delone sets are rectifiable

We show that every linearly repetitive Delone set in the Euclidean d-space Rd, with d⩾2, is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice Zd. In the particular case when the Delone set X in Rd comes from a primitive substitution tiling of Rd, we give a condition on the eigenvalues of the substitution matrix which ensures the existence of a homeomorphism with bounded displacement from X to the lattice βZd for some positive β. This condition includes primitive Pisot substitution tilings but also concerns a much broader set of substitution tilings.

Multi-Type Directed Scale-Free Percolation

In this paper, we study a long-range percolation model on the lattice ℤd with multi-type vertices and directed edges. Each vertex x ∊ ℤd is independently assigned a non-negative weight Wx and a type ψx, where (Wx)x∊ℤd are i.i.d. random variables, and (ψx)x∊ℤd are also i.i.d. Conditionally on weights and types, and given λ, α > 0, the edges are independent and the probability that there is a directed edge from x to y is given by pxy = 1 − exp(−λφψxψy WxWy/|x−y|α), where φij's are entries from a type matrix Φ. We show that, when the tail of the distribution of Wx is regularly varying with exponent τ − 1, the tails of the out/in-degree distributions are both regularly varying with exponent γ = α(τ − 1)/d. We formulate conditions under which there exist critical values λcWCC ∊ (0, ∞) and λcSCC ∊ (0, ∞) such that an infinite weak component and an infinite strong component emerge, respectively, when λ exceeds them. A phase transition is established for the shortest path lengths of directed and undirected edges in the infinite component at the point γ = 2, where the out/in-degrees switch from having finite to infinite variances. The random graph model studied here features some structures of multi-type vertices and directed edges which appear naturally in many real-world networks, such as the SNS networks and computer communication networks.