Hamiltonians on discrete structures: jumps of the integrated density of states and uniform convergence

We study equivariant families of discrete Hamiltonians on amenable geometries and their integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spanned by eigenfunctions with compact support. The size of a jump of the IDS is consequently given by the equivariant dimension of the subspace spanned by such eigenfunctions. From this we deduce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants of the IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and random operators on quasi-transitive graphs, and operators on percolation graphs.     

The Wegner estimate and the integrated density of states for some random operators

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local H?lder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on theL ^p-theory of the spectral shift function (SSF), forp ≥ 1, applicable to pairs of self-adjoint operators whose difference is in the trace idealI _p, for 0p ≤ 1. We present this and other results on the SSF due to other authors. Under an additional condition of the single-site potential, local H?lder continuity is proved at all energies. Finally, we present extensions of this work to random potentials with nonsign definite single-site potentials.     

A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states

In this paper we consider bounded operators on infinite graphs, in particular Cayley graphs of amenable groups. The operators satisfy an equivariance condition which is formulated in terms of a colouring of the vertex set of the underlying graph. In this setting it is natural to expect that the integrated density of states (IDS), or spectral distribution function, exists. We show that it can be defined as the uniform limit of approximants associated to finite matrices. The proof is based on a Banach space valued ergodic theorem which even allows explicit convergence estimates. Our result applies to a variety of group structures and colouring types, in particular to periodic operators and percolation-type Hamiltonians.     

Global Continuity of the Integrated Density of States for Random Landau Hamiltonians

Abstract

We prove that the integrated density of states (IDS) for the randomly perturbed Landau Hamiltonian is Hölder continuous at all energies with any Hölder exponent 0 < q < 1. The random Anderson-type potential is constructed with a nonnegative, compactly supported single-site potential u. The distribution of the iid random variables is required to be absolutely continuous with a bounded, compactly supported density. This extends a previous result Combes et al. [Combes, J. M., Hislop, P. D., Klopp, F. (2003a). Hölder continuity of the integrated density of states for some random operators at all energies. Int. Math. Res. Notices 2003: 179--209] that was restricted to constant magnetic fields having rational flux through the unit square. We also prove that the IDS is Hölder continuous as a function of the nonzero magnetic field strength.     

The Integrated Density of States for Some Random Operators with Nonsign Definite Potentials

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L^p-theory of the spectral shift function for p?1 (Comm. Math. Phys.218 (2001), 113-130), and the vector field methods of Klopp (Comm. Math. Phys.167 (1995), 553-569). We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.     

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

We establish quantitative homogenization, large‐scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well‐approximated by harmonic functions on ℝ^d. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher‐order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville‐type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|^k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil‐Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ. © 2018 Wiley Periodicals, Inc.     

Bootstrap percolation with inhibition

We study a variant of the classical bootstrap percolation process on Erd?s Rényi random graphs. The graphs we consider have inhibitory vertices obstructing the diffusion of activity and excitatory vertices facilitating it. We study both a synchronous and an asynchronous version of the process. Both begin with a small initial set of active vertices, and the activation spreads to all vertices for which the number of excitatory active neighbors exceeds the number of inhibitory active neighbors by a certain amount. We show that in the synchronous process, inhibitory vertices may cause unstable behavior: tiny changes in the size of the starting set can dramatically influence the size of the final active set. We further show that in the asynchronous model the process becomes stable and stops with an active set containing a nontrivial deterministic constant fraction of all vertices. Moreover, we show that percolation occurs significantly faster asynchronously than synchronously.     

A density approach to Hamilton-Jacobi equations with t-measurable Hamiltonians

In 1985 H. Ishii [Is85] proposed a generalization of the notion of (continuous) viscosity solution for an Hamilton-Jacobi equation with a t-measurable Hamiltonian—that is, a Hamiltonian which is measurable in time and continuous in the other variables. This notion turned out to agree with natural applications, like Control and Differential Games Theory. Since then, several improvements have been achieved for the standard situation when the Hamiltonian is continuous. It is someway an accepted general idea that parallel improvements are likely for t-measurable Hamiltonians as well, though such a job might appear a bit tedious because of the necessarily involved technicalities.In this paper we show that Ishii’s definition of viscosity solution coincides with the one which would arise by extending by density the standard definition. Namely, we regard a t-measurable Hamiltonian H as an element of the closure (for suitable topologies) of a class of continuous Hamiltonians. On the other hand, we show that the set of Ishii’s (sub-, super-) solutions for H is nothing but the limit set of the (sub-, super-) solutions corresponding to continuous Hamiltonians approaching H. This put us in the condition of establishing comparison, existence, and regularity results by deriving them from the analogous results for the case of continuous Hamiltonians.     

Uniform existence of the integrated density of states for models on $${\mathbb{Z}}^d$$

We provide an ergodic theorem for certain Banach-space valued functions on structures over , which allow for existence of frequencies of finite patterns. As an application we obtain existence of the integrated density of states for associated discrete finite-range operators in the sense of convergence of the distributions with respect to the supremum norm. These results apply to various examples including periodic operators, percolation models and nearest-neighbour hopping on the set of visible points. Our method gives explicit bounds on the speed of convergence in terms of the speed of convergence of the underlying frequencies. It uses neither von Neumann algebras nor a framework of random operators on a probability space.      

Nearest neighbor and hard sphere models in continuum percolation

Consider a Poisson process X in R ^d with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). © 1996 John Wiley & Sons, Inc.     

Comportement à très basses énergies de la densité d'états intégrée du modèle d'Anderson non borné inférieurement

We obtain a complete asymptotic expansion of the integrated density of states of the unbounded Anderson model at low energies. We also study the evolution of this asymptotic when the decay of the tail of the distribution of the random potential increases. To cite this article: O. Saad, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper, we study unimodular amenable groups. The first part of the paper is devoted to results on the existence of uniform families of ε-quasi tilings for these groups. First we extend constructions of Ornstein and Weiss by quantitative estimates for the covering properties of the corresponding decompositions. Then we apply the methods developed to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable amenable group into some Banach space. This result extends significantly and complements related results found in the literature. Further, using the Lindenstrauss ergodic theorem, we link our results to classical ergodic theory. We conclude with two important applications: uniform approximation of the integrated density of states on amenable Cayley graphs and almost-sure convergence of cluster densities in an amenable bond percolation model.     

Sharpness versus robustness of the percolation transition in 2d contact processes

We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in Kéfi et al. (2007) for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of ‘robust critical behaviour’ with power law cluster size distribution for a range of parameter values, as suggested in Kéfi et al. (2007).     

Absolutely continuous spectrum for random Schrödinger operators on the Bethe strip

The Bethe strip of width m is the cartesian product $\mathbb {B}\times \lbrace 1,\ldots ,m\rbrace$, where $\mathbb {B}$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have “extended states” for small disorder. More precisely, we consider Anderson‐like Hamiltonians $H_\lambda =\frac{1}{2} \Delta \otimes 1 + 1 \otimes A\,+\,\lambda \mathcal {V}$ on a Bethe strip with connectivity K ≥ 2, where A is an m × m symmetric matrix, $\mathcal {V}$ is a random matrix potential, and λ is the disorder parameter. Given any closed interval $I\subset \big (\!-\!\sqrt{K}+a_{{\rm max}},\sqrt{K}+a_{\rm {min}}\big )$, where a_min and a_max are the smallest and largest eigenvalues of the matrix A, we prove that for λ small the random Schrödinger operator H_λ has purely absolutely continuous spectrum in I with probability one and its integrated density of states is continuously differentiable on the interval I.     

Uniform existence of the integrated density of states for random Schrödinger operators on metric graphs over

We consider ergodic random Schrödinger operators on the metric graph with random potentials and random boundary conditions taking values in a finite set. We show that normalized finite volume eigenvalue counting functions converge to a limit uniformly in the energy variable. This limit, the integrated density of states, can be expressed by a closed Shubin–Pastur type trace formula. It supports the spectrum and its points of discontinuity are characterized by existence of compactly supported eigenfunctions. Among other examples we discuss random magnetic fields and percolation models.     

Percolation on random recursive trees

We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree‐indexed process of cluster sizes to the genealogical tree of a continuous‐state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous‐time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 655–680, 2016     

Continuum percolation with steps in the square or the disc

In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density 1, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area A_c for percolation in this model. The proof is in two parts: First we give a rigorous reduction of the problem to a finite problem, and then we solve this problem using Monte‐Carlo methods. We prove that, with 99.99% confidence, the critical area lies between 4.508 and 4.515. For the corresponding problem with the disc replaced by the square we prove, again with 99.99% confidence, that the critical area lies between 4.392 and 4.398. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

The number of infinite clusters in dynamical percolation

Summary. Dynamical percolation is a Markov process on the space of subgraphs of a given graph, that has the usual percolation measure as its stationary distribution. In previous work with O. H?ggstr?m, we found conditions for existence of infinite clusters at exceptional times. Here we show that for ℤ ^d , with p>p _c , a.s. simultaneously for all times there is a unique infinite cluster, and the density of this cluster is θ(p). For dynamical percolation on a general tree Γ, we show that for p>p _c , a.s. there are infinitely many infinite clusters at all times. At the critical value p=p _c , the number of infinite clusters may vary, and exhibits surprisingly rich behaviour. For spherically symmetric trees, we find the Hausdorff dimension of the set T ^k of times where the number of infinite clusters is k, and obtain sharp capacity criteria for a given time set to intersect T ^k . The proof of this capacity criterion is based on a new kernel truncation technique. Received: 5 May 1997 / In revised form: 24 November 1997     

On the variational description of the trajectories of averaging quantum dynamical maps

We study the dynamics of quantum system with degenerated Hamiltonian. To this end we consider the approximating sequence of regularized Hamiltonians and corresponding sequence of dynamical semigroups acting in the space of quantum states. The limit points set of the sequence of regularized semigroups is obtained as the result of averaging by finitely additive measure on the set of regularizing parameters. We establish that the family of averaging dynamical maps does not possess the semigroup property and the injectivity property. We define the functionals on the space of maps of the time interval into the quantum states space such that the maximum points of this functionals coincide with the trajectories of the family of averaging dynamical maps.     

Periodic Operators in High-Contrast Media and the Integrated Density of States Function

ABSTRACT

This paper studies the asymptotic behavior for the integrated density of states function for operators associated with the propagation of classical waves in a high-contrast, periodic, two-component medium. Consider a domain Ω₊ contained in the hypercube [0, 2π) ⁿ . We define a function χ_τ which takes the value 1 in Ω₊ and the value τ in [0, 2π) ⁿ \ Ω₊. We extend this setup periodically to ? ⁿ and define the operator L _τ = ??χ_τ ?. As τ goes to infinity, it is known that the spectrum of L _τ exhibits a band-gap structure and that the spectral density accumulates at the upper endpoints of the bands. We establish the existence and some important properties of a rescaled integrated density of states function in the large coupling limit which describes the non-trivial asymptotic behavior of this spectral accumulation.