Lattice Zd Research Articles

2×2 operator matrix with real parameter and its spectrum

In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix Aμ (so called generalized Friedrichs model) with real parameter μ ∈ R. It is associated with the Hamiltonian of a system consisting of at most two particles on a d -dimensional lattice Zd, interacting via creation and annihilation operators. Aμ is linear bounded self-adjoint operator acting in the two-particle cut subspace of the Fock space, that is, in the direct sum of zero-particle and one-particle subspaces of a Fock space. We find the essential and discrete spectra of the block operator matrix Aμ. The Fredholm determinant and resolvent operator associated to Aμ are constructed. The spectrum of Aμ plays an important role in the study of the spectral properties of the Hamiltonians associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles on a lattice.

Invariance principle for the capacity and the cardinality of the range of stable random walks

We prove an almost sure invariance principle for the capacity and the cardinality of the range of a class of α-stable random walks on the integer lattice Zd with d/α>5/2, and d/α>3/2, respectively. As a direct consequence, we conclude Khintchine’s and Chung’s laws of the iterated logarithm for both processes.

A variational formula for large deviations in first-passage percolation under tail estimates

Consider the first passage percolation on the d-dimensional lattice Zd with identical and independent weight distributions and the first passage time T. In this paper, we study the upper tail large deviations P(T(0,nx)>n(μ+ξ)), for ξ>0 and x≠0 with a time constant μ, for weights that satisfy a tail assumption P(τe>t)≍βexp(−αtr). When r≤1 (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as exp(−(2dαξr+o(1))n). When 1<r≤d, we find that the rate function can be naturally described by a variational formula, called the discrete p-Capacity, and we study its asymptotics. The case r=d is critical and logarithmic corrections appear. For r∈(1,d), we show that the large deviation event {T(0,nx)>n(μ+ξ)} is described by a localization of high weights around the endpoints. The picture changes for r≥d where the configuration is not anymore localized.

Delone sets that are not rectifiable under Lipschitz co-uniformly continuous bijections

We prove that there exist Delone sets in Rd, d≥2, which cannot be mapped onto the standard lattice Zd by Lipschitz co-uniformly continuous bijections satisfying an asymptotic control on the lower distortion. The impossibility of the unrectifiability crucially uses ideas of Lipschitz regular maps recently introduced by M. Dymond, V. Kaluža and E. Kopecká.

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers of products of d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a d-dimensional discrete torus Gm which depends on m. The sums in question are then related to the nth step of a Markov chain on Gm. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice Zd which covers Gm.

Percolation of worms

We introduce a new correlated percolation model on the d-dimensional lattice Zd called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and v∈(0,∞) (the intensity parameter). From each site of Zd we start POI(v) independent simple random walks with this length distribution. We investigate the connectivity properties of the set Sv of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then Sv undergoes a percolation phase transition as v varies. Our main contribution is a sufficient condition on the length distribution which guarantees that Sv percolates for all v>0 if d≥5. E.g., if the probability mass function of the length distribution is m(ℓ)=c⋅ln(ln(ℓ))ɛ/(ℓ3ln(ℓ))1[ℓ≥ℓ0]for some ℓ0>ee and ɛ>0 then Sv percolates for all v>0. Note that the second moment of this length distribution is only “barely” infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Bernoulli hyper-edge percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being “extremal” in this family of models.

Homogenization of discrete thin structures

We consider graphs parameterized on a portion X⊂Zd×{1,…,M}k of a cylindrical subset of the lattice Zd×Zk, and perform a discrete-to-continuum dimension-reduction process for energies defined on X of quadratic type. Our only assumptions are that X be connected as a graph and periodic in the first d-directions. We show that, upon scaling of the domain and of the energies by a small parameter ɛ, the scaled energies converge to a d-dimensional limit energy. The main technical points are a dimension-reducing coarse-graining process and a discrete version of the p-connectedness approach by Zhikov.

An elliptic Harnack inequality for difference equations with random balanced coefficients

We prove an elliptic Harnack inequality at large scale on the lattice Zd for nonnegative solutions of a difference equation with balanced i.i.d. coefficients which are not necessarily elliptic. We also identify the optimal constant in the Harnack inequality. Our proof relies on a quantitative homogenization result of the corresponding invariance principle to Brownian motion and on percolation estimates. As a corollary of our main theorem, we derive an almost optimal Hölder estimate.

Entropy decay in the Swendsen–Wang dynamics on Zd

We study the mixing time of the Swendsen–Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice Zd. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is nonlocal, that is, it changes the entire configuration in one step. We prove that, whenever strong spatial mixing (SSM) holds, the mixing time on any n-vertex cube in Zd is O(logn), and we prove this is tight by establishing a matching lower bound on the mixing time. The previous best known bound was O(n). SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in d=2 dimensions throughout the high-temperature (single phase) region. Our result follows from a modified log-Sobolev inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen–Wang dynamics, which extends to general bipartite graphs of bounded degree. This factorization leads to several additional results, including mixing time bounds for a number of natural local and nonlocal Markov chains on the joint space, as well as for the standard random-cluster dynamics.

Spectral dimension of simple random walk on a long-range percolation cluster

Consider the long-range percolation model on the integer lattice Zd in which all nearest-neighbour edges are present and otherwise x and y are connected with probability qx,y:=1−exp(−|x−y|−s), independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e. for s>d,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value d=1, s=2, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters s∈(d,2d). We further note that our approach is applicable to short-range models as well.

Maximal edge-traversal time in First-passage percolation

In this paper, we study the maximal edge-traversal time on optimal paths in First-passage percolation on the lattice Zd for several edge distributions, including the Pareto and Weibull distributions. It is known to be unbounded when the edge distribution has unbounded support [J. van den Berg and H. Kesten. Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 56-80, 1993]. We determine the order of the growth depending on the tail of the edge distribution.

Biased random walk on supercritical percolation: anomalous fluctuations in the ballistic regime

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice Zd for d≥2. For this model, Fribergh and Hammond showed the existence of an exponent γ such that: for γ<1, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by γ; whereas for γ>1, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when γ>2. In this article, we explain how Fribergh and Hammond’s percolation estimates further allow it to be established that for γ∈(1,2) the fluctuations about the mean are of an anomalous polynomial order, with exponent given by γ−1.

Bound states of Schrödinger-type operators on one and two dimensional lattices

We study the spectral properties of the Schrödinger-type operatorHˆμ:=Hˆ0+μVˆ,μ≥0, associated to a one-particle system in d-dimensional lattice Zd, d=1,2, where the non-perturbed operator Hˆ0 is a self-adjoint convolution-type operator generated by a Hopping matrix eˆ:Zd→C and the potential Vˆ is the multiplication operator by vˆ:Zd→R. Under certain regularity assumption on eˆ and a decay assumption on vˆ, we establish the existence or non-existence and also the finiteness of eigenvalues of Hˆμ. Moreover, in the case of existence we study the asymptotics of eigenvalues of Hˆμ as μ↘0.

Cut-off for sandpiles on tiling graphs

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling, either torus or open boundary conditions. A general method of obtaining the Green’s function of the tiling is given, and a total variation cut-off phenomenon is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings. In a companion paper, computational methods are used to demonstrate that an open boundary condition alters the mixing time for the D4 lattice in dimension 4, while an asymptotic evaluation shows that it does not change the asymptotic mixing time for the cubic lattice Zd for all sufficiently large d.

On the monotonicity of the critical time in the Constrained-degree percolation model

The Constrained-degree percolation model was introduced in de Lima et al. (2020), where it was proven that this model has a non-trivial phase transition on a square lattice. We study the Constrained-degree percolation model on the d-dimensional hypercubic lattice (Zd) and, via numerical simulations, found evidence that the critical time tcd(k) is monotonous not increasing in the constrained k if d=3,4, like it is when d=2. We verify that the lowest constrained value k such that the system exhibits a phase transition is k=3 and that the correlation critical exponent ν for the Constrained-degree percolation model and ordinary Bernoulli percolation are the same.

Heavy-tailed branching random walks on multidimensional lattices. A moment approach

We study a continuous-time branching random walk (BRW) on the lattice ℤd, d ∈ ℕ, with a single source of branching, that is the lattice point where the birth and death of particles can occur. The random walk is assumed to be spatially homogeneous, symmetric and irreducible but, in contrast to the majority of previous investigations, the random walk transition intensities a(x, y) decrease as |y − x|−(d+α) for |y − x| → ∞, where α ∈ (0, 2), that leads to an infinite variance of the random walk jumps. The mechanism of the birth and death of particles at the source is governed by a continuous-time Markov branching process. The source intensity is characterized by a certain parameter β. We calculate the long-time asymptotic behaviour for all integer moments for the number of particles at each lattice point and for the total population size. With respect to the parameter β, a non-trivial critical point βc &gt; 0 is found for every d ≥ 1. In particular, if β &gt; βc the evolutionary operator generated a behaviour of the first moment for the number of particles has a positive eigenvalue. The existence of a positive eigenvalue yields an exponential growth in t of the particle numbers in the case β &gt; βc called supercritical. Classification of the BRW treated as subcritical (β &lt; βc) or critical (β = βc) for the heavy-tailed random walk jumps is more complicated than for a random walk with a finite variance of jumps. We study the asymptotic behaviour of all integer moments of a number of particles at any point y ∈ ℤd and of the particle population on ℤd according to the ratio d/α.

Autocovariance varieties of moving average random fields

We study the autocovariance functions of moving average random fields over the integer lattice Zd from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.

Hydrodynamics of the weakly asymmetric normalized binary contact path process

We are concerned with the weakly asymmetric normalized binary contact path process. The symmetric version was introduced in Griffeath (1983). The model describes the spread of an infectious disease on the lattice Zd. The configuration at each site x∈Zd takes value in [0,∞). Site x is recovered at rate 1, and is infected by its neighbor x±ei at rate λ∓λ1∕N, where {ei}1≤i≤d is the canonical basis of the d-dimensional lattice, λ,λ1 are non-negative constants and N is the scaling parameter. When the infection occurs, the seriousness of the disease at site x is added with that of y. When there is neither recovery nor infection occurring during some time interval, the value at site x evolves according to some ODE. We prove that for d≥3 and large value λ, the empirical measure of the process, under diffusive scaling, converges in probability to a deterministic measure whose density is the unique weak solution to a linear parabolic equation, while for small λ and in all dimensions, the limit is zero. We also conjecture that the fluctuations field in the former case is driven by a generalized Ornstein–Uhlenbeck process, while a rigorous proof is absent. The main difficulty in proving the hydrodynamics is to prove the absolute continuity of the limiting path, where the theory of linear systems introduced in Liggett (1985) is utilized.

Asymptotic Behavior of the Mean Number of Particles for a Branching Random Walk on the Lattice Zd with Periodic Sources of Branching

We consider a continuous-time branching random walk on ℤ d with birth and death of particles at a periodic set of points (sources of branching). Spectral properties of the evolution operator of the mean number of particles are studied. We derive a representation of the mean value of particle number in a form of asymptotic series.

Transport of a quantum particle in a time-dependent white-noise potential

We show that a quantum particle in Rd, for d ⩾ 1, subject to a white-noise potential, moves superballistically in the sense that the mean square displacement ∫∥x∥2⟨ρ(x, x, t)⟩dx grows like t3 in any dimension. The white-noise potential is Gaussian distributed with an arbitrary spatial correlation function and a delta correlation function in time. Similar results were established in one dimension by Jayannavar and Kumar [Phys. Rev. Lett. 48(8), 553–556 (1982)], and for any dimension using different methods by Fischer et al. [Phys. Rev. Lett. 73(12), 1578–1581 (1994)]. We also prove that for the same white-noise potential model on the lattice Zd, for d ⩾ 1, the mean square displacement is diffusive growing like t1. This behavior on the lattice is consistent with the diffusive behavior observed for similar models on the lattice Zd with a time-dependent Markovian potential by Kang and Schenker [J. Stat. Phys. 134, 1005–1022 (2009)].