Abstract: Abstract
The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses ≤ 1. The process stabilizes almost surely if m < 1 and it almost surely does not stabilize if m > 1, where m is the mean mass per vertex. The main result of this paper is that in the critical case m = 1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete bi-Laplacian Gaussian field. PubDate: 2016-07-01

Abstract: Abstract
We consider translation invariant gapped quantum spin systems satisfying the Lieb–Robinson bound and containing single-particle states in a ground state representation. Following the Haag–Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding S-matrix. We also obtain some general restrictions on the shape of the energy–momentum spectrum. For the purpose of our analysis, we adapt the concepts of almost local observables and energy–momentum transfer (or Arveson spectrum) from relativistic QFT to the lattice setting. The Lieb–Robinson bound, which is the crucial substitute of strict locality from relativistic QFT, underlies all our constructions. Our results hold, in particular, in the Ising model in strong transverse magnetic fields. PubDate: 2016-07-01

Abstract: Abstract
We place the hyperbolic quantum Ruijsenaars–Schneider system with an exponential Morse term on a lattice and diagonalize the resulting n-particle model by means of multivariate continuous dual q-Hahn polynomials that arise as a parameter reduction of the Macdonald–Koornwinder polynomials. This allows to compute the n-particle scattering operator, to identify the bispectral dual system, and to confirm the quantum integrability in a Hilbert space setup. PubDate: 2016-07-01

Abstract: Abstract
We consider commuting operators obtained by quantization of Hamiltonians of the Hopf (aka dispersionless KdV) hierarchy. Such operators naturally arise in the setting of Symplectic Field Theory (SFT). A complete set of common eigenvectors of these operators is given by Schur polynomials. We use this result for computing the SFT potential of a disk. PubDate: 2016-07-01

Abstract: Abstract
We consider the Kuramoto model of globally coupled phase oscillators in its continuum limit, with individual frequencies drawn from a distribution with density of class
\({C^n}\)
(
\({n\geq 4}\)
). A criterion for linear stability of the uniform stationary state is established which, for basic examples in the literature, is equivalent to the standard condition on the coupling strength. We prove that, under this criterion, the Kuramoto order parameter, when evolved under the full nonlinear dynamics, asymptotically vanishes (with polynomial rate n) for every trajectory issued from a sufficiently small
\({C^n}\)
perturbation. The proof uses techniques from the Analysis of PDEs and closely follows recent proofs of the nonlinear Landau damping in the Vlasov equation and Vlasov-HMF model. PubDate: 2016-07-01

Abstract: Abstract
We consider the discrete spectrum of the two-dimensional Hamiltonian H = H
0 + V, where H
0 is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V, we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H. PubDate: 2016-07-01

Abstract: Abstract
We construct an inductive system of C*-algebras each of which is isomorphic to a finite tensor product of copies of the one-mode n-th degree polynomial extension of the usual Weyl algebra constructed in our previous paper (Accardi and Dhahri in Open Syst Inf Dyn 22(3):1550001, 2015). We prove that the inductive limit C*-algebra is factorizable and has a natural localization given by a family of C*-sub-algebras each of which is localized on a bounded Borel subset of
\({\mathbb{R}}\)
. Finally, we prove that the corresponding family of Fock states, defined on the inductive family of C*-algebras, is projective if and only if n = 1. This is a weak form of the no-go theorems which emerge in the study of representations of current algebras over Lie algebras. PubDate: 2016-07-01

Abstract: Abstract
We consider a finite region of a d-dimensional lattice,
\({d \in \mathbb{N}}\)
, of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size
\({\varepsilon}\)
. Each oscillator weakly interacts by force of order
\({\varepsilon}\)
with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as
\({\varepsilon \rightarrow 0}\)
behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order
\({\varepsilon^{-1}}\)
and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next, we assume that the interaction potential is of size
\({\varepsilon \lambda}\)
, where
\({\lambda}\)
is another small parameter, independent from
\({\varepsilon}\)
. Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit
\({\varepsilon \rightarrow 0}\)
, the main order in
\({\lambda}\)
of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space–time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system. PubDate: 2016-07-01

Abstract: Abstract
We show that a suitable choice of the initial data from the past null infinity of Christodoulou’s work on the formation of black holes (The formation of black holes in general relativity. Monographs in Mathematics, European Mathematical Society, Zürich, 2009) will lead to the formation of a near Schwarzschild region up to the past null infinity. PubDate: 2016-07-01

Abstract: Abstract
We show that the Anderson model has a transition from localization to delocalization at exactly two-dimensional growth rate on antitrees with normalized edge weights which are certain discrete graphs. The kinetic part has a one-dimensional structure allowing a description through transfer matrices which involve some Schur complement. For such operators we introduce the notion of having one propagating channel and extend theorems from the theory of one-dimensional Jacobi operators that relate the behavior of transfer matrices with the spectrum. These theorems are then applied to the considered model. In essence, in a certain energy region the kinetic part averages the random potentials along shells and the transfer matrices behave similar as for a one-dimensional operator with random potential of decaying variance. At d-dimensional growth for
\({d > 2}\)
this effective decay is strong enough to obtain absolutely continuous spectrum, whereas for some uniform d-dimensional growth with
\({d < 2}\)
one has pure point spectrum in this energy region. At exactly uniform two-dimensional growth also some singular continuous spectrum appears, at least at small disorder. As a corollary we also obtain a change from singular spectrum (
\({d \leq 2}\)
) to absolutely continuous spectrum (
\({d \geq 3)}\)
for random operators of the type
\({\mathcal{P}_r\Delta_d\mathcal{P}_r + \lambda\mathcal{V}}\)
on
\({\mathbb{Z}^d}\)
, where
\({\mathcal{P}_r}\)
is an orthogonal radial projection,
\({\Delta_d}\)
the discrete adjacency operator (Laplacian) on
\({\mathbb{Z}^d}\)
and
\({\lambda\mathcal{V}}\)
a random potential. PubDate: 2016-07-01

Abstract: Abstract
We give an exhaustive description of bifurcations and of the number of solutions of the vacuum Lichnerowicz equation with positive cosmological constant on
\({S^1\times S^2}\)
with
\({U(1)\times SO(3)}\)
-invariant seed data. The resulting CMC slicings of Schwarzschild–de Sitter and Nariai are described. PubDate: 2016-06-17

Abstract: Abstract
We consider an instanton, A, with L
2-bounded curvature F
A on the cylindrical manifold
\({Z=\mathbf{R}
\times M}\)
, where M is a closed Riemannian n-manifold,
\({n \geq 4}\)
. We assume M admits a smooth 3-form P and a smooth 4-form Q satisfy
\({dP=4Q}\)
and
\({d\ast_{M}{Q}=(n-3)\ast_{M}P}\)
. Manifolds with these forms include nearly Kähler 6-manifolds and nearly parallel G
2-manifolds in dimension 7. Then we can prove that the instanton must be a flat connection. PubDate: 2016-06-17

Abstract: Abstract
We study the four-dimensional n-component
\({ \varphi ^4}\)
spin model for all integers
\({n \ge 1}\)
and the four-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case
\({n=0}\)
interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power
\({\frac 12\frac{n+2}{n+8}}\)
, for all
\({n \ge 0}\)
and
\({p > 0}\)
. The proof is based on an improvement of a rigorous renormalisation group method developed previously. PubDate: 2016-06-15

Abstract: Abstract
The Holstein model has widely been accepted as a model comprising electrons interacting with phonons; analysis of this model’s ground states was accomplished two decades ago. However, the results were obtained without completely taking repulsive Coulomb interactions into account. Recent progress has made it possible to treat such interactions rigorously; in this paper, we study the Holstein–Hubbard model with repulsive Coulomb interactions. The ground-state properties of the model are investigated; in particular, the ground state of the Hamiltonian is proven to be unique for an even number of electrons on a bipartite connected lattice. In addition, we provide a rigorous upper bound on charge susceptibility. PubDate: 2016-06-15

Abstract: Abstract
We prove a local trace formula for Anosov flows. It relates Pollicott–Ruelle resonances to the periods of closed orbits. As an application, we show that the counting function for resonances in a sufficiently wide strip cannot have a sublinear growth. In particular, for any Anosov flow there exist strips with infinitely many resonances. PubDate: 2016-06-14

Abstract: Abstract
In the small noise regime, the average transition time between metastable states of a reversible diffusion process is described at the logarithmic scale by Arrhenius’ law. The Eyring–Kramers formula classically provides a subexponential prefactor to this large deviation estimate. For irreversible diffusion processes, the equivalent of Arrhenius’ law is given by the Freidlin–Wentzell theory. In this paper, we compute the associated prefactor and thereby generalise the Eyring–Kramers formula to irreversible diffusion processes. In our formula, the role of the potential is played by Freidlin–Wentzell’s quasipotential, and a correction depending on the non-Gibbsianness of the system along the minimum action paths is highlighted. Our study assumes some properties for the vector field: (1) attractors are isolated points, (2) the dynamics restricted to basin of attraction boundaries are attracted to single points (which are saddle-points of the vector field). We moreover assume that the minimum action paths that connect attractors to adjacent saddle-points (the instantons) have generic properties that are summarised in the conclusion. At a technical level, our derivation combines an exact computation for the first-order WKB expansion around the instanton and an exact computation of the first-order match asymptotics expansion close to the saddle-point. While the results are exact once a formal expansion is assumed, the validity of these asymptotic expansions remains to be proven. PubDate: 2016-06-11

Abstract: Abstract
We discuss spectral properties of a charged quantum particle confined to a chain graph consisting of an infinite array of rings under the influence of a magnetic field assuming a
\({\delta}\)
-coupling at the points where the rings touch. We start with the situation when the system has a translational symmetry and analyze spectral consequences of perturbations of various kind, such as a local change of the magnetic field, of the coupling constant, or of a ring circumference. A particular attention is paid to weak perturbations, both local and periodic; for the later, we prove a version of Saxon–Hutner conjecture. PubDate: 2016-06-11

Abstract: Abstract
We consider the Cauchy problem for the nonlinear Schrödinger equations of fractional order
$$\left\{\begin{array}{l}i\partial _{t}u-2\left( -\partial _{x}^{2}
\right)^{\frac{1}{4}} \, u=F\left( u\right) \\ u\left( 0,x\right) =u_{0}
\left( x\right),\end{array}\right.$$
where
\({F\left( u\right) }\)
is the cubic nonlinearity
$$F\left( u\right) =\lambda \left u\right ^{2}u$$
with
\({\lambda \in \mathbf{R}}\)
. We find the large time asymptotics of solutions to the Cauchy problem. We use the factorization technique similar to that developed for the Schrödinger equation. PubDate: 2016-06-10

Abstract: Abstract
Recently we introduced T-duality in the study of topological insulators, and used it to show that T-duality transforms the bulk–boundary homomorphism into a simpler restriction map in two dimensions. In this paper, we partially generalize these results to higher dimensions in both the complex and real cases, and briefly discuss the 4D quantum Hall effect. PubDate: 2016-06-10

Abstract: Abstract
We present lower bounds for the uniform radius of spatial analyticity of solutions to the Korteweg–de Vries equation, which improve earlier results due to Bona, Grujić and Kalisch. PubDate: 2016-06-09