Lattice Zd Research Articles

On resonances in disordered multi-particle systems

We assess the probability of resonances between sufficiently distant states x=(x1,…,xN) and y=(y1,…,yN) in the configuration space of an N-particle disordered quantum system on the lattice Zd, d⩾1. This includes the cases where the transition x⇝y “shuffles” the particles in x, like the transition (a,a,b)⇝(a,b,b) in a 3-particle system. In presence of a random external potential V(⋅,ω) such pairs of configurations (x,y) give rise to strongly coupled random local Hamiltonians, so that eigenvalue concentration bounds are difficult to obtain (cf. Aizenman and Warzel (2009) [2]; Chulaevsky and Suhov (2009) [8]). This results in eigenfunction decay bounds weaker than expected. We show that more optimal bounds obtained so far only for 2-particle systems (Chulaevsky and Suhov (2008) [6]) can be extended to any N>2.

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators Hγμ(k), k ∈ $\mathbb{T}^d $ := (−π,π]d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤd, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of Hγμ(k), k ∈ $\mathbb{T}^d $ , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ $\mathbb{T}^d $ . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ $\mathbb{T}^d $ in the existence domain G ⊂ $\mathbb{T}^d $ .

Spectral estimates for Schrödinger operators with sparse potentials on graphs

A construction of potentials, suggested by the authors for the lattice ℤd, d gt; 2, is extended to a large class of combinatorial and metric graphs whose global dimension is a number D gt; 2. For the Schrodinger operator - Δ - αV on such graphs, with a sparse potential V, we study the behavior (as α → ∞) of the number N_(-Δ - αV) of negative eigenvalues of - Δ - αV. We show that by means of sparse potentials one can realize any prescribed asymptotic behavior of N_(-Δ - αV) under very mild regularity assumptions. A similar construction works also for the lattice ℤ2, where D = 2.

Disordered Fermions on Lattices and Their Spectral Properties

We study Fermionic systems on a lattice with random interactions through their dynamics and the associated KMS states. These systems require a more complex approach compared with the standard spin systems on a lattice, on account of the difference in commutation rules for the local algebras for disjoint regions, between these two systems. It is for this reason that some of the known formulations and proofs in the case of the spin lattice systems with random interactions do not automatically go over to the case of disordered Fermion lattice systems. We extend to the disordered CAR algebra some standard results concerning the spectral properties exhibited by temperature states of disordered quantum spin systems. We investigate the Arveson spectrum, known to physicists as the set of the Bohr frequencies. We also establish its connection with the Connes and Borchers spectra, and with the associated invariants for such W ∗-dynamical systems which determine the type of von Neumann algebras generated by a temperature state. We prove that all such spectra are independent of the disorder. Such results cover infinite-volume limits of finite-volume Gibbs states, that is the quenched disorder for Fermions living on a standard lattice ℤ d , including models exhibiting some standard spin-glass-like behavior. As a natural application, we show that a temperature state can generate only a type $\mathop {\rm {III}}$ von Neumann algebra (with the type $\mathop {\rm {III_{0}}}$ component excluded). In the case of the pure thermodynamic phase, the associated von Neumann algebra is of type $\mathop {\rm {III_{\lambda }}}$ for some λ∈(0,1], independent of the disorder. All such results are in accordance with the principle of self-averaging which affirms that the physically relevant quantities do not depend on the disorder. The approach pursued in the present paper can be viewed as a further step towards fully understanding the very complicated structure of the set of temperature states of quantum spin glasses, and its connection with the breakdown of the symmetry for the replicas.

A model of continuous time polymer on the lattice

In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice Zd, and the environment is a collection {W (t, x); t ≥ 0, x ∈ Zd} of i.i.d. Brownian motions.

Translation-Equivariant Matchings of Coin Flips on ℤd

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤdand is given by a deterministic function of the coin flips. LetZΦbe the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour ofZΦfor translation-equivariant perfect matching rules? We prove that, for everyd≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε< ∞ for every ε > 0.

Translation-Equivariant Matchings of Coin Flips on ℤd

Consider independent fair coin flips at each site of the lattice ℤd. A translation-equivariant matching rule is a perfect matching of heads to tails that commutes with translations of ℤd and is given by a deterministic function of the coin flips. Let ZΦ be the distance from the origin to its partner, under the translation-equivariant matching rule Φ. Holroyd and Peres (2005) asked, what is the optimal tail behaviour of ZΦ for translation-equivariant perfect matching rules? We prove that, for every d ≥ 2, there exists a translation-equivariant perfect matching rule Φ such that EZΦ2/3-ε < ∞ for every ε > 0.

Activated Random Walkers: Facts, Conjectures and Challenges

We study a particle system with hopping (random walk) dynamics on the integer lattice ℤd. The particles can exist in two states, active or inactive (sleeping); only the former can hop. The dynamics conserves the number of particles; there is no limit on the number of particles at a given site. Isolated active particles fall asleep at rate λ>0, and then remain asleep until joined by another particle at the same site. The state in which all particles are inactive is absorbing. Whether activity continues at long times depends on the relation between the particle density ζ and the sleeping rate λ. We discuss the general case, and then, for the one-dimensional totally asymmetric case, study the phase transition between an active phase (for sufficiently large particle densities and/or small λ) and an absorbing one. We also present arguments regarding the asymptotic mean hopping velocity in the active phase, the rate of fixation in the absorbing phase, and survival of the infinite system at criticality. Using mean-field theory and Monte Carlo simulation, we locate the phase boundary. The phase transition appears to be continuous in both the symmetric and asymmetric versions of the process, but the critical behavior is very different. The former case is characterized by simple integer or rational values for critical exponents (β=1, for example), and the phase diagram is in accord with the prediction of mean-field theory. We present evidence that the symmetric version belongs to the universality class of conserved stochastic sandpiles, also known as conserved directed percolation. Simulations also reveal an interesting transient phenomenon of damped oscillations in the activity density.

Finite Trotter Approximation to the Averaged Mean Square Distance in the Anderson Model

We prove that a finite Trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice ℤd exhibits at most a diffusive behavior in dimensions d≥3 as long as the Fourier transform of the single-site probability, \(\hat{\mu }\), is in L2(ℝ).

Error Analysis of Frame Reconstruction From Noisy Samples

This paper addresses the problem of reconstructing a continuous function defined on Rd from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean zero and variance sigma2. We sample the continuous function / on the uniform lattice (1/m)Zd, and show for large enough m that the variance of the error between the frame reconstruction fepsiv,m from noisy samples of f and the function f satisfy var(fepsiv,m(x) - f(x)) ap(sigma2/md)Cx where Cx is the best constant for every x isin Rd. We also prove a similar result in the case that our data are weighted-average samples of / corrupted by additive noise.

Asymptotic Decay of Correlations for a Random Walk on the Lattice Zd in Interaction with a Markov Field

We consider a discrete-time random walk on Z d , d = 1;2;::: in a random environment with Markov evolution in time. We complete and extend to all dimension d ‚ 1 the results obtained in [2] on the time decay of the correlations of the \environment from the point of view of the random walk. Math. Subj. Class.: 60J15, 82B10, 82B41.

Positivity of the two-particle Hamiltonian on a lattice

We consider a two-particle Hamiltonian on the d-dimensional lattice ℤd. We find a sufficient condition for the positivity of a family of operators h(k) appearing after the “separation of the center of mass” of a system of two particles depending on the values of the total quasimomentum k ∈ Td (where Td is a d-dimensional torus). We use the obtained result to show that the operator h(k) has positive eigenvalues for nonzero k ∈ Td.

LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS

We study nonlinear dispersive wave systems described by hyperbolic PDE's in ℝdand difference equations on the lattice ℤd. The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi–Pasta–Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully-developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

On approximate pattern matching for a class of Gibbs random fields

We prove an exponential approximation for the law of approximate occurrence of typical patterns for a class of Gibssian sources on the lattice ℤd, d≥2. From this result, we deduce a law of large numbers and a large deviation result for the waiting time of distorted patterns.

Mass extinctions: an alternative to the Allee effect

We introduce a spatial stochastic process on the lattice Zd to model mass extinctions. Each site of the lattice may host a flock of up to N individuals. Each individual may give birth to a new individual at the same site at rate ϕ until the maximum of N individuals has been reached at the site. Once the flock reaches N individuals, then, and only then, it starts giving birth on each of the 2d neighboring sites at rate λ(N). Finally, disaster strikes at rate 1, that is, the whole flock disappears. Our model shows that, at least in theory, there is a critical maximum flock size above which a species is certain to disappear and below which it may survive.

The Randomized Integer Convex Hull

Let K ⊂ Rd be a sufficiently round convex body (the ratio of the circumscribed ball to the inscribed ball is bounded by a constant) of a sufficiently large volume. We investigate the randomized integer convex hull IL(K) = conv (K ⋂ L), where L is a randomly translated and rotated copy of the integer lattice Zd. We estimate the expected number of vertices of IL(K), whose behaviour is similar to the expected number of vertices of the convex hull of Vol K random points in K. In the planar case we also describe the expectation of the missed area Vol (K \ IL(K)). Surprisingly, for K a polygon, the behaviour in this case is different from the convex hull of random points.

Rigorous Analysis of Discontinuous Phase Transitions via Mean-Field Bounds

We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice ℤ d . Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean- field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on ℤ d undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q≥3, the r-component cubic models with r≥4 and the O(N)-nematic liquid-crystal models with N≥3.

The Minesweeper Game: Percolation and Complexity

We study a model motivated by the minesweeper game. In this model one starts with the percolation of mines on the lattice ℤd, and then tries to find an infinite path of mine-free sites. At every recovery of a free site, the player is given some information on the sites adjacent to the current site. We compare the parameter values for which there exists a strategy such that the process survives to the critical parameter of ordinary percolation. We then derive improved bounds for these values for the same process, when the player has some complexity restrictions in computing his moves. Finally, we discuss some monotonicity issues which arise naturally for this model.

Statistics for the contact process

A d‐dimensional contact process is a simplified model for the spread of an infection on the lattice Zd. At any given time t≥0, certain sites x∈Zd are infected while the remaining once are healthy. Infected sites recover at constant rate 1, while healthy sites are infected at a rate proportional to the number of infected neighboring sites. The model is parametrized by the proportionality constant λ. If λ is sufficiently small, infection dies out (subcritical process), whereas if λ is sufficiently large infection tends to be permanent (supercritical process).In this paper we study the estimation problem for the parameter λ of the supercritical contact process starting with a single infected site at the origin. Based on an observation of this process at a single time t, we obtain an estimator for the parameter λ which is consistent and asymptotically normal as t→∞

On Effective Conductivity on Z d Lattice

We study the effective conductivity σe for a random wire problem on the d-dimensional cubic lattice ℤd, d≥2 in the case when random conductivities on bonds are independent identically distributed random variables. We give exact expressions for the expansion of the effective conductivity in terms of the moments of the disorder parameter up to the 5th order. In the 2D case using the duality symmetry we also derive the 6th order expansion. We compare our results with the Bruggeman approximation and show that in the 2D case it coincides with the exact solution up to the terms of 4th order but deviates from it for the higher order terms.